Introduction
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Limit from Definition
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- Definition of Limit
- Definition of Limit - Examples
- One-Sided Limit
- Infinite Limit
- Limit at Infinity
- Definition of Continuity
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20

Technique 1 Substitution
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Technique 2 Factoring
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Technique 3 Multiplying by the Conjugate
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Technique 4 Function Tends to Infinity
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Technique 5 X Tends to Infinity
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- X Tends to Infinity Part 1
- X Tends to Infinity Part 2
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23

Technique 6 Eulers Limit
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Technique 7 Trigonometric Limits
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Technique 8 The Sandwich Squeeze Theorem
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Technique 9 Piecewise Functions
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Introduction","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Limit of a Function","Duration":"10m 15s","ChapterTopicVideoID":8248,"CourseChapterTopicPlaylistID":161,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this clip, I\u0027ll be introducing the limit of a function."},{"Start":"00:04.380 ","End":"00:07.695","Text":"The concept of a limit is so important,"},{"Start":"00:07.695 ","End":"00:11.985","Text":"it\u0027s considered to be the basis of the differential and integral calculus."},{"Start":"00:11.985 ","End":"00:14.280","Text":"If you look ahead at the exercises,"},{"Start":"00:14.280 ","End":"00:17.310","Text":"you will see examples such as this,"},{"Start":"00:17.310 ","End":"00:19.980","Text":"which will look very strange to you if you don\u0027t know,"},{"Start":"00:19.980 ","End":"00:22.235","Text":"if you\u0027ve never seen this symbol."},{"Start":"00:22.235 ","End":"00:24.045","Text":"It\u0027s actually short for limit."},{"Start":"00:24.045 ","End":"00:28.170","Text":"In short, this will all look like gibberish and you won\u0027t know what\u0027s expected from you."},{"Start":"00:28.170 ","End":"00:31.935","Text":"Hopefully, by the end of this introduction, it\u0027ll be clearer."},{"Start":"00:31.935 ","End":"00:34.230","Text":"Meanwhile, let me get this out of your site."},{"Start":"00:34.230 ","End":"00:37.990","Text":"Let\u0027s begin by examining the function that appears here,"},{"Start":"00:37.990 ","End":"00:43.430","Text":"f of x is equal to x squared minus 1 over x minus 1."},{"Start":"00:43.430 ","End":"00:46.940","Text":"Its domain is clearly x not equal to"},{"Start":"00:46.940 ","End":"00:50.620","Text":"1 because that\u0027s the only value of x that could make it go wrong,"},{"Start":"00:50.620 ","End":"00:52.115","Text":"so let me write that."},{"Start":"00:52.115 ","End":"00:56.555","Text":"The domain is any x except x equals 1."},{"Start":"00:56.555 ","End":"00:58.580","Text":"If you try to substitute x equals 1,"},{"Start":"00:58.580 ","End":"01:01.430","Text":"you get 0 over 0 and that\u0027s not defined."},{"Start":"01:01.430 ","End":"01:03.905","Text":"But if I can\u0027t substitute x equals 1,"},{"Start":"01:03.905 ","End":"01:06.080","Text":"I could ask the following question."},{"Start":"01:06.080 ","End":"01:11.750","Text":"What happens to y or what does y approach when x approaches 1?"},{"Start":"01:11.750 ","End":"01:13.460","Text":"Let me try to explain."},{"Start":"01:13.460 ","End":"01:16.625","Text":"We\u0027re not allowed to substitute x equals 1."},{"Start":"01:16.625 ","End":"01:18.290","Text":"If I substitute x equals 1,"},{"Start":"01:18.290 ","End":"01:20.495","Text":"I don\u0027t have a value of y."},{"Start":"01:20.495 ","End":"01:24.200","Text":"But what if x just gets close to 1,"},{"Start":"01:24.200 ","End":"01:28.730","Text":"say x equals 0.99 or closer,"},{"Start":"01:28.730 ","End":"01:34.555","Text":"does y approach 4 maybe 10, 100 minus 1?"},{"Start":"01:34.555 ","End":"01:40.640","Text":"What happens here when x approaches a value where the function is undefined?"},{"Start":"01:40.640 ","End":"01:42.890","Text":"This end, we\u0027ll make a table of x,"},{"Start":"01:42.890 ","End":"01:46.235","Text":"y values and try to make an educated guess."},{"Start":"01:46.235 ","End":"01:47.765","Text":"Let\u0027s put some values in."},{"Start":"01:47.765 ","End":"01:50.250","Text":"We said x approaches 1."},{"Start":"01:50.250 ","End":"01:53.940","Text":"Let\u0027s begin with x equals 1.1."},{"Start":"01:53.940 ","End":"01:55.605","Text":"I\u0027ll be your calculator,"},{"Start":"01:55.605 ","End":"01:57.110","Text":"I\u0027ll substituted in here,"},{"Start":"01:57.110 ","End":"01:59.760","Text":"and y equals 2.1."},{"Start":"01:59.760 ","End":"02:03.245","Text":"Let\u0027s get even closer to 1, and let\u0027s try 1.01."},{"Start":"02:03.245 ","End":"02:06.260","Text":"Again. I\u0027ll be your calculator and I\u0027ll tell"},{"Start":"02:06.260 ","End":"02:09.740","Text":"you that what we get if we substitute 1.01 for x,"},{"Start":"02:09.740 ","End":"02:12.755","Text":"we\u0027ll get 2.01 for y."},{"Start":"02:12.755 ","End":"02:14.689","Text":"Let\u0027s get still closer,"},{"Start":"02:14.689 ","End":"02:21.195","Text":"1.001, you get 2.001."},{"Start":"02:21.195 ","End":"02:22.875","Text":"Let\u0027s do 1 more."},{"Start":"02:22.875 ","End":"02:26.355","Text":"When x equals 1.0001,"},{"Start":"02:26.355 ","End":"02:28.740","Text":"that\u0027s pretty close to 1,"},{"Start":"02:28.740 ","End":"02:33.180","Text":"do the computation, y equals 2.0001."},{"Start":"02:33.180 ","End":"02:39.830","Text":"All this leaves us with a distinct impression that when x approaches 1, y approaches 2."},{"Start":"02:39.830 ","End":"02:42.740","Text":"But something here is not quite right or not fair."},{"Start":"02:42.740 ","End":"02:45.380","Text":"All these values have been larger than 1."},{"Start":"02:45.380 ","End":"02:46.985","Text":"We\u0027re getting close to 1,"},{"Start":"02:46.985 ","End":"02:49.130","Text":"but we\u0027re staying above 1."},{"Start":"02:49.130 ","End":"02:51.500","Text":"We should really do it from the other side too,"},{"Start":"02:51.500 ","End":"02:54.920","Text":"and start with x equals say 0.9,"},{"Start":"02:54.920 ","End":"02:58.570","Text":"and approaches 1 from the other side. Let\u0027s do that."},{"Start":"02:58.570 ","End":"03:04.010","Text":"Let\u0027s start a second table and again we have x here and y here."},{"Start":"03:04.010 ","End":"03:07.610","Text":"This time we\u0027ll start from below 1 and work our way up."},{"Start":"03:07.610 ","End":"03:13.890","Text":"If x is 0.9, then y equals 1.9."},{"Start":"03:13.890 ","End":"03:16.235","Text":"That\u0027s for the rest, I\u0027ll just keep writing them in."},{"Start":"03:16.235 ","End":"03:22.510","Text":"If x equals 0.99, y equals 1.99."},{"Start":"03:22.510 ","End":"03:28.965","Text":"If we let x equals still closer 0.999, 1.999,"},{"Start":"03:28.965 ","End":"03:32.280","Text":"and the last 0.9999,"},{"Start":"03:32.280 ","End":"03:36.370","Text":"we get the value of y, 1.9999."},{"Start":"03:36.370 ","End":"03:40.790","Text":"We once again see that the values of y get very close to 2,"},{"Start":"03:40.790 ","End":"03:42.425","Text":"just as they did here."},{"Start":"03:42.425 ","End":"03:46.190","Text":"This allows us with some confidence to write the following."},{"Start":"03:46.190 ","End":"03:51.460","Text":"The Limit, which is written lim, just abbreviated,"},{"Start":"03:51.460 ","End":"04:00.605","Text":"as x goes to 1 of x squared minus 1 over x minus 1 is equal to 2."},{"Start":"04:00.605 ","End":"04:04.895","Text":"This is the notation lim for limit and this arrow,"},{"Start":"04:04.895 ","End":"04:06.230","Text":"x goes to 1,"},{"Start":"04:06.230 ","End":"04:08.930","Text":"x tends to 1, x approaches 1."},{"Start":"04:08.930 ","End":"04:10.730","Text":"That\u0027s the notation for limits."},{"Start":"04:10.730 ","End":"04:14.630","Text":"There\u0027s a comment I\u0027d like to make before we continue with the main theme."},{"Start":"04:14.630 ","End":"04:20.015","Text":"If I just want to describe what I concluded from this lower table,"},{"Start":"04:20.015 ","End":"04:25.100","Text":"where x approach 1 from below from numbers smaller than 1,"},{"Start":"04:25.100 ","End":"04:27.875","Text":"0.9999, and so on,"},{"Start":"04:27.875 ","End":"04:30.365","Text":"there is a special notation for that."},{"Start":"04:30.365 ","End":"04:38.510","Text":"We write x approaches 1 from the left and we write the little minus above the 1."},{"Start":"04:38.510 ","End":"04:41.145","Text":"This does not mean minus 1,"},{"Start":"04:41.145 ","End":"04:44.190","Text":"it just means slightly less than 1,"},{"Start":"04:44.190 ","End":"04:46.520","Text":"like 0.999 and so on."},{"Start":"04:46.520 ","End":"04:51.050","Text":"This describes this table and this sometimes we say the limit from the left."},{"Start":"04:51.050 ","End":"04:55.250","Text":"That the limit as x goes to 1 from the left of this and this is 2."},{"Start":"04:55.250 ","End":"04:58.010","Text":"Notice that I don\u0027t also put a minus above the 2,"},{"Start":"04:58.010 ","End":"05:00.485","Text":"even though it\u0027s slightly less than 2,"},{"Start":"05:00.485 ","End":"05:01.985","Text":"that\u0027s not of interest."},{"Start":"05:01.985 ","End":"05:04.805","Text":"Similarly, in the table above,"},{"Start":"05:04.805 ","End":"05:08.090","Text":"we had x approach 1 from numbers larger than 1,"},{"Start":"05:08.090 ","End":"05:10.820","Text":"1.1, 1.0, 1 etc."},{"Start":"05:10.820 ","End":"05:15.720","Text":"To describe this, we write a little plus above the 1,"},{"Start":"05:15.720 ","End":"05:19.340","Text":"and we say x approaches 1 from the right,"},{"Start":"05:19.340 ","End":"05:23.725","Text":"or the limit as x approaches 1 from the right of this function is 2."},{"Start":"05:23.725 ","End":"05:26.450","Text":"Again, we don\u0027t write that plus over the 2."},{"Start":"05:26.450 ","End":"05:29.840","Text":"These 2 are called the 1-sided limits,"},{"Start":"05:29.840 ","End":"05:32.900","Text":"the limit from the right and the limit from the left."},{"Start":"05:32.900 ","End":"05:35.060","Text":"We do have terms for these 2,"},{"Start":"05:35.060 ","End":"05:38.540","Text":"and this is just generally the limit from either side."},{"Start":"05:38.540 ","End":"05:43.190","Text":"Note that after I saw that the limit from above,"},{"Start":"05:43.190 ","End":"05:47.390","Text":"from the right was 2 and the limit from the left was 2,"},{"Start":"05:47.390 ","End":"05:50.810","Text":"then I had to write 2-sided limit as this is called"},{"Start":"05:50.810 ","End":"05:54.560","Text":"or just the plain limit was also equal to 2."},{"Start":"05:54.560 ","End":"05:57.050","Text":"Let\u0027s go on to the next page."},{"Start":"05:57.050 ","End":"06:00.020","Text":"To summarize this, I saw that limit from"},{"Start":"06:00.020 ","End":"06:03.300","Text":"the right was equal to the limit of the left which is equal to 2,"},{"Start":"06:03.300 ","End":"06:07.550","Text":"and then I wrote that the regular limit or 2-sided limit was equal to 2."},{"Start":"06:07.550 ","End":"06:10.535","Text":"Now, in most of our exercises,"},{"Start":"06:10.535 ","End":"06:13.190","Text":"we don\u0027t actually need to bother with limit"},{"Start":"06:13.190 ","End":"06:16.050","Text":"from the right and limit from the left and show that their equal."},{"Start":"06:16.050 ","End":"06:19.190","Text":"In most cases we just have techniques and we find the limit."},{"Start":"06:19.190 ","End":"06:22.250","Text":"There is 1 exceptional case where we will be dealing"},{"Start":"06:22.250 ","End":"06:25.600","Text":"with limit from the left separately and then add from the right separately,"},{"Start":"06:25.600 ","End":"06:27.980","Text":"and only when we conclude that these 2 are equal,"},{"Start":"06:27.980 ","End":"06:29.765","Text":"then we do the 2-sided limit."},{"Start":"06:29.765 ","End":"06:33.095","Text":"These are not so common, but you still have to know about this."},{"Start":"06:33.095 ","End":"06:35.990","Text":"What we learned here can actually be summarized"},{"Start":"06:35.990 ","End":"06:39.200","Text":"in a theorem which I\u0027m going to write below."},{"Start":"06:39.200 ","End":"06:42.360","Text":"Theorem, a function has a limit at a point,"},{"Start":"06:42.360 ","End":"06:43.460","Text":"and In our case,"},{"Start":"06:43.460 ","End":"06:46.280","Text":"this is the function and this is the point,"},{"Start":"06:46.280 ","End":"06:47.310","Text":"if and only if,"},{"Start":"06:47.310 ","End":"06:50.750","Text":"this is how in mathematics we say if and only if,"},{"Start":"06:50.750 ","End":"06:54.365","Text":"it has a limit from the left and from the right of the point,"},{"Start":"06:54.365 ","End":"06:56.530","Text":"and if these 2 are equal."},{"Start":"06:56.530 ","End":"06:59.090","Text":"Actually, I should have added something else that if"},{"Start":"06:59.090 ","End":"07:01.580","Text":"all of this happens and in this case,"},{"Start":"07:01.580 ","End":"07:03.830","Text":"that\u0027s the value of the limit of the function."},{"Start":"07:03.830 ","End":"07:06.950","Text":"For example, here this was 2 and this was 2,"},{"Start":"07:06.950 ","End":"07:08.705","Text":"so this limit not only exists,"},{"Start":"07:08.705 ","End":"07:10.115","Text":"but it\u0027s also equal to 2,"},{"Start":"07:10.115 ","End":"07:11.450","Text":"but I think that\u0027s implied."},{"Start":"07:11.450 ","End":"07:15.065","Text":"Before we continue to the actual techniques for finding limits,"},{"Start":"07:15.065 ","End":"07:19.495","Text":"I\u0027d like to illustrate all this graphically and they\u0027ll begin on the next page."},{"Start":"07:19.495 ","End":"07:22.325","Text":"I present the graph of our function."},{"Start":"07:22.325 ","End":"07:25.415","Text":"Note, especially this hole here."},{"Start":"07:25.415 ","End":"07:27.740","Text":"In general, this comes out to be a straight line,"},{"Start":"07:27.740 ","End":"07:29.360","Text":"but with a missing point,"},{"Start":"07:29.360 ","End":"07:32.150","Text":"because when x equals 1,"},{"Start":"07:32.150 ","End":"07:33.710","Text":"there is no value of y,"},{"Start":"07:33.710 ","End":"07:35.075","Text":"so I left a gap."},{"Start":"07:35.075 ","End":"07:41.865","Text":"But it certainly confirms the fact that when x is near 1 and y comes out near 2."},{"Start":"07:41.865 ","End":"07:46.370","Text":"What we did before was we just took values of x above 1,"},{"Start":"07:46.370 ","End":"07:52.460","Text":"getting closer, maybe some point here, and then here."},{"Start":"07:52.460 ","End":"07:55.890","Text":"Then we basically said, okay."},{"Start":"07:55.890 ","End":"08:00.065","Text":"That\u0027s x going to 1 from the right."},{"Start":"08:00.065 ","End":"08:02.780","Text":"Then we took points like the 0.99,"},{"Start":"08:02.780 ","End":"08:06.389","Text":"but various points that were on this side,"},{"Start":"08:06.550 ","End":"08:13.180","Text":"and this side, and that took care of the direction of coming to 1 from the left."},{"Start":"08:13.180 ","End":"08:19.395","Text":"This was the 1 minus and the other 1 was the 1 plus, let\u0027s call it."},{"Start":"08:19.395 ","End":"08:25.520","Text":"We computed the values of y and we got points on the graph when x is 1, there is no y."},{"Start":"08:25.520 ","End":"08:27.890","Text":"Nevertheless, if x is very close to 1,"},{"Start":"08:27.890 ","End":"08:32.000","Text":"y is very close to 2 and this line is 2 and this is 1."},{"Start":"08:32.000 ","End":"08:33.920","Text":"A big problem though,"},{"Start":"08:33.920 ","End":"08:37.010","Text":"is the fact that this is so informal, this process."},{"Start":"08:37.010 ","End":"08:39.050","Text":"What do I mean just taking a table,"},{"Start":"08:39.050 ","End":"08:40.370","Text":"putting in a few values,"},{"Start":"08:40.370 ","End":"08:42.200","Text":"and guessing what the trend is?"},{"Start":"08:42.200 ","End":"08:43.670","Text":"That just simply won\u0027t do."},{"Start":"08:43.670 ","End":"08:45.490","Text":"It\u0027s not mathematically enough."},{"Start":"08:45.490 ","End":"08:49.880","Text":"I\u0027d like to show you more algebraic way of getting to this value 2."},{"Start":"08:49.880 ","End":"08:53.600","Text":"I mean, we got ultimately that the conclusion that the limit was 2,"},{"Start":"08:53.600 ","End":"08:55.070","Text":"just from 2 tables."},{"Start":"08:55.070 ","End":"08:57.155","Text":"From the table, from the right table from the left,"},{"Start":"08:57.155 ","End":"08:59.810","Text":"there were equal, the numbers looked like they were getting close to 2."},{"Start":"08:59.810 ","End":"09:01.969","Text":"Now, let me do something more precise."},{"Start":"09:01.969 ","End":"09:04.310","Text":"Here\u0027s how I would do this algebraically."},{"Start":"09:04.310 ","End":"09:08.270","Text":"I would say the limit as x goes to 1,"},{"Start":"09:08.270 ","End":"09:09.845","Text":"this is the original question,"},{"Start":"09:09.845 ","End":"09:14.540","Text":"x squared minus 1 over x minus 1 is equal 2."},{"Start":"09:14.540 ","End":"09:18.755","Text":"What I would first of all do is some algebraic formula."},{"Start":"09:18.755 ","End":"09:21.310","Text":"But there\u0027s a thing called difference of squares formula."},{"Start":"09:21.310 ","End":"09:25.910","Text":"In any event, you must have seen this countless times that x squared minus 1 can be"},{"Start":"09:25.910 ","End":"09:32.695","Text":"factorized as 2 x minus 1 x plus 1 over x minus 1."},{"Start":"09:32.695 ","End":"09:36.140","Text":"It\u0027s agreed that when we take a limit where x tends to 1,"},{"Start":"09:36.140 ","End":"09:38.410","Text":"we do not include the value 1."},{"Start":"09:38.410 ","End":"09:40.565","Text":"We\u0027re not dividing by 0,"},{"Start":"09:40.565 ","End":"09:42.200","Text":"I mean x is from our domain."},{"Start":"09:42.200 ","End":"09:43.580","Text":"Is not equal to 1,"},{"Start":"09:43.580 ","End":"09:46.670","Text":"I mean, and x minus 1 is not 0."},{"Start":"09:46.670 ","End":"09:49.010","Text":"Therefore, just by the rules of fractions,"},{"Start":"09:49.010 ","End":"09:50.960","Text":"were allowed to cancel something in"},{"Start":"09:50.960 ","End":"09:53.855","Text":"the numerator and in the denominator or a common factor."},{"Start":"09:53.855 ","End":"09:56.370","Text":"What we\u0027re left here is just x plus 1."},{"Start":"09:56.370 ","End":"10:02.440","Text":"Now 1 of the first techniques we\u0027ll learn in the limits is method of substitution."},{"Start":"10:02.440 ","End":"10:05.720","Text":"If there\u0027s no particular problem and x plus 1 is not problematic,"},{"Start":"10:05.720 ","End":"10:07.430","Text":"we just substitute the value."},{"Start":"10:07.430 ","End":"10:10.340","Text":"You just put x equals 1 into here."},{"Start":"10:10.340 ","End":"10:11.870","Text":"We got to the answer 2,"},{"Start":"10:11.870 ","End":"10:13.430","Text":"which we were expecting."},{"Start":"10:13.430 ","End":"10:16.140","Text":"This will do as an introduction."}],"ID":8408}],"Thumbnail":null,"ID":161},{"Name":"Limit from Definition","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Definition of Limit","Duration":"12m 28s","ChapterTopicVideoID":8151,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.740","Text":"In this clip, we\u0027re going to give"},{"Start":"00:01.740 ","End":"00:06.600","Text":"a more formal definition of the limit of a function and I\u0027m assuming"},{"Start":"00:06.600 ","End":"00:09.480","Text":"you have some practical experience with the limit of"},{"Start":"00:09.480 ","End":"00:12.885","Text":"a function and an intuitive knowledge of what it means."},{"Start":"00:12.885 ","End":"00:16.770","Text":"We\u0027re just going to make it more formal and sometimes"},{"Start":"00:16.770 ","End":"00:20.740","Text":"people refer to this as the epsilon-delta definition."},{"Start":"00:20.740 ","End":"00:23.735","Text":"Let\u0027s first of all specify the setup."},{"Start":"00:23.735 ","End":"00:26.940","Text":"We have a function f,"},{"Start":"00:26.940 ","End":"00:30.015","Text":"let\u0027s say f of the variable x,"},{"Start":"00:30.015 ","End":"00:34.790","Text":"and I want it to be defined on"},{"Start":"00:34.790 ","End":"00:41.870","Text":"an interval containing x equals a."},{"Start":"00:41.870 ","End":"00:43.520","Text":"It could be defined for all x,"},{"Start":"00:43.520 ","End":"00:46.350","Text":"but at least on some interval containing a."},{"Start":"00:46.690 ","End":"00:54.215","Text":"What we\u0027re going to do is to define symbol that we\u0027ve used a lot."},{"Start":"00:54.215 ","End":"00:59.105","Text":"We\u0027re going to define what it means that the limit as x goes to"},{"Start":"00:59.105 ","End":"01:05.440","Text":"a or tends to a of f of x is equal to some number L,"},{"Start":"01:05.440 ","End":"01:07.005","Text":"L for limit,"},{"Start":"01:07.005 ","End":"01:11.995","Text":"and I\u0027m going to do it both formally and informally."},{"Start":"01:11.995 ","End":"01:16.310","Text":"Let me write a formal definition first of all."},{"Start":"01:17.420 ","End":"01:22.510","Text":"I\u0027m going to do it formally and then afterwards,"},{"Start":"01:22.510 ","End":"01:25.235","Text":"I\u0027ll give you informally"},{"Start":"01:25.235 ","End":"01:29.725","Text":"because I want you to understand it and not just to have it precise."},{"Start":"01:29.725 ","End":"01:33.400","Text":"Before that, there is something I forgot to add"},{"Start":"01:33.400 ","End":"01:37.420","Text":"that it doesn\u0027t have to be defined at a itself."},{"Start":"01:37.420 ","End":"01:45.290","Text":"f is defined on an interval containing a and I\u0027ll write except possibly at a."},{"Start":"01:45.860 ","End":"01:50.020","Text":"It might or might not be defined at x equals a,"},{"Start":"01:50.020 ","End":"01:52.715","Text":"the point is that we don\u0027t care."},{"Start":"01:52.715 ","End":"01:57.295","Text":"I\u0027ll give these 2 definitions and I\u0027ll also highlight it."},{"Start":"01:57.295 ","End":"02:00.475","Text":"This is what we\u0027re going to define in 2 ways."},{"Start":"02:00.475 ","End":"02:09.000","Text":"Formally, this means that for every number epsilon,"},{"Start":"02:09.000 ","End":"02:12.840","Text":"this is the Greek letter epsilon, little epsilon."},{"Start":"02:12.840 ","End":"02:16.840","Text":"It\u0027s like an e, has to be bigger than 0."},{"Start":"02:18.500 ","End":"02:20.865","Text":"For each epsilon,"},{"Start":"02:20.865 ","End":"02:26.240","Text":"there is a delta also bigger than"},{"Start":"02:26.240 ","End":"02:31.880","Text":"0 and actually,"},{"Start":"02:31.880 ","End":"02:33.470","Text":"delta depends on epsilon."},{"Start":"02:33.470 ","End":"02:35.120","Text":"Each epsilon will give us a different delta."},{"Start":"02:35.120 ","End":"02:42.180","Text":"So sometimes we write it as delta of epsilon instead of just plain delta,"},{"Start":"02:42.180 ","End":"02:45.705","Text":"but we understand that each epsilon has its own delta."},{"Start":"02:45.705 ","End":"02:49.050","Text":"There it is. Now I need a such that,"},{"Start":"02:49.050 ","End":"02:59.844","Text":"such that the modulus or absolute value of f of x"},{"Start":"02:59.844 ","End":"03:05.335","Text":"minus L is less than epsilon"},{"Start":"03:05.335 ","End":"03:12.120","Text":"whenever the absolute value"},{"Start":"03:12.120 ","End":"03:20.405","Text":"of x minus a is less than delta."},{"Start":"03:20.405 ","End":"03:25.400","Text":"Notice that this means that f of x is close to L or"},{"Start":"03:25.400 ","End":"03:29.550","Text":"at least is close within an error of epsilon."},{"Start":"03:29.550 ","End":"03:33.115","Text":"So it\u0027s between L plus epsilon and L minus epsilon."},{"Start":"03:33.115 ","End":"03:37.130","Text":"Similarly here, x is going to be close to a within"},{"Start":"03:37.130 ","End":"03:43.380","Text":"delta and that means it\u0027s between a plus delta and a minus delta,"},{"Start":"03:43.380 ","End":"03:48.260","Text":"but this is not quite right because we don\u0027t"},{"Start":"03:48.260 ","End":"03:54.320","Text":"know that the function is defined at x equals a itself."},{"Start":"03:54.320 ","End":"04:03.035","Text":"So what we can do is just adjust this slightly by saying this has to be bigger than 0."},{"Start":"04:03.035 ","End":"04:08.475","Text":"Bigger than 0 for absolute value just means that it\u0027s not 0,"},{"Start":"04:08.475 ","End":"04:11.760","Text":"this just means that x is not equal to a."},{"Start":"04:11.760 ","End":"04:15.120","Text":"That\u0027s the formal definition."},{"Start":"04:15.120 ","End":"04:19.040","Text":"I don\u0027t expect you to understand or digest it at this point,"},{"Start":"04:19.040 ","End":"04:20.855","Text":"I just want to have it written down."},{"Start":"04:20.855 ","End":"04:23.720","Text":"I\u0027m going to write an informal definition and then we show how"},{"Start":"04:23.720 ","End":"04:26.660","Text":"the 2 tie in and then how we work with it,"},{"Start":"04:26.660 ","End":"04:28.695","Text":"and hopefully, in time,"},{"Start":"04:28.695 ","End":"04:31.900","Text":"you\u0027ll get an intuition of what this all means."},{"Start":"04:31.900 ","End":"04:35.155","Text":"Informally, I\u0027ll say this,"},{"Start":"04:35.155 ","End":"04:44.935","Text":"that we can make f of x as close as we like to L by"},{"Start":"04:44.935 ","End":"04:50.390","Text":"making x close enough"},{"Start":"04:50.390 ","End":"04:57.815","Text":"to a. I have to add but not equal to a,"},{"Start":"04:57.815 ","End":"05:02.720","Text":"close enough to a but not equal to"},{"Start":"05:02.720 ","End":"05:08.690","Text":"a because we said we don\u0027t know what happens at x equals a."},{"Start":"05:08.690 ","End":"05:12.995","Text":"We don\u0027t even know if the function\u0027s defined at x equals a."},{"Start":"05:12.995 ","End":"05:16.285","Text":"Making f as close as we like to L,"},{"Start":"05:16.285 ","End":"05:19.285","Text":"we think of epsilon like an error,"},{"Start":"05:19.285 ","End":"05:23.620","Text":"how much error we\u0027ll tolerate so that if I give you an epsilon,"},{"Start":"05:23.620 ","End":"05:28.930","Text":"I mean I want it within epsilon of L. The epsilon could be very small,"},{"Start":"05:28.930 ","End":"05:32.425","Text":"could be 0.001, whatever."},{"Start":"05:32.425 ","End":"05:36.130","Text":"I can get f of x close to L within"},{"Start":"05:36.130 ","End":"05:42.790","Text":"that small error provided that x is close enough to a,"},{"Start":"05:42.790 ","End":"05:46.240","Text":"which means finding a delta."},{"Start":"05:46.240 ","End":"05:48.910","Text":"We\u0027ll call this 1 the tolerance."},{"Start":"05:48.910 ","End":"05:53.740","Text":"So whenever x is within delta of a,"},{"Start":"05:53.740 ","End":"05:57.730","Text":"then we\u0027re guaranteed that f of x is within that margin"},{"Start":"05:57.730 ","End":"06:02.750","Text":"of error around L. A picture will help."},{"Start":"06:03.110 ","End":"06:07.505","Text":"Here\u0027s the graph of some function f of x."},{"Start":"06:07.505 ","End":"06:11.045","Text":"This is the y-axis and this is the x-axis."},{"Start":"06:11.045 ","End":"06:15.500","Text":"What we\u0027re concerned is what happens around x equals a."},{"Start":"06:15.500 ","End":"06:23.170","Text":"We don\u0027t know what happens when x equals a. I\u0027ll indicate that with a red circle here."},{"Start":"06:23.170 ","End":"06:24.995","Text":"It could be defined here,"},{"Start":"06:24.995 ","End":"06:26.420","Text":"might not be defined,"},{"Start":"06:26.420 ","End":"06:29.825","Text":"it could be defined as something else completely."},{"Start":"06:29.825 ","End":"06:33.170","Text":"We don\u0027t actually care what happens at x equals a,"},{"Start":"06:33.170 ","End":"06:36.395","Text":"we care what happens when x is close to a,"},{"Start":"06:36.395 ","End":"06:39.005","Text":"close but not equal to."},{"Start":"06:39.005 ","End":"06:41.140","Text":"What happens is this,"},{"Start":"06:41.140 ","End":"06:45.605","Text":"we want to make sure that when x is very close to a,"},{"Start":"06:45.605 ","End":"06:50.010","Text":"that f of x is very close to L. It\u0027s like a game."},{"Start":"06:50.010 ","End":"06:51.900","Text":"Someone gives you an epsilon,"},{"Start":"06:51.900 ","End":"06:58.205","Text":"which is like a margin of error and says I want f of x to be within epsilon of"},{"Start":"06:58.205 ","End":"07:06.980","Text":"L. Within epsilon is like within this horizontal band and I can see in this picture,"},{"Start":"07:06.980 ","End":"07:14.235","Text":"in this particular case, that when x is within delta of a,"},{"Start":"07:14.235 ","End":"07:16.800","Text":"then this part of the diagram,"},{"Start":"07:16.800 ","End":"07:19.530","Text":"I\u0027ll highlight it so you see what I mean,"},{"Start":"07:19.530 ","End":"07:23.840","Text":"this part here is the part where"},{"Start":"07:23.840 ","End":"07:29.180","Text":"x is between a plus or minus delta and we can see that the y,"},{"Start":"07:29.180 ","End":"07:34.235","Text":"the f of x is completely within the horizontal strip."},{"Start":"07:34.235 ","End":"07:37.835","Text":"There\u0027s even some room to spare at the top."},{"Start":"07:37.835 ","End":"07:42.605","Text":"So this delta is certainly good for this epsilon."},{"Start":"07:42.605 ","End":"07:47.285","Text":"Epsilon can be as small as anyone cares to give us,"},{"Start":"07:47.285 ","End":"07:48.800","Text":"could be 1 over a million,"},{"Start":"07:48.800 ","End":"07:50.270","Text":"1 over a billion."},{"Start":"07:50.270 ","End":"07:52.460","Text":"No matter how small epsilon is,"},{"Start":"07:52.460 ","End":"07:54.305","Text":"if someone gives us an epsilon,"},{"Start":"07:54.305 ","End":"07:57.600","Text":"we can make parallel lines,"},{"Start":"07:57.600 ","End":"08:06.190","Text":"we can go to the curve and we can find some vertical strip that will completely fit,"},{"Start":"08:06.190 ","End":"08:11.435","Text":"the function intersects that vertical strip completely within the horizontal strip."},{"Start":"08:11.435 ","End":"08:15.425","Text":"In time, the idea will become clear."},{"Start":"08:15.425 ","End":"08:19.010","Text":"Perhaps I\u0027ll also show you something else,"},{"Start":"08:19.010 ","End":"08:23.795","Text":"a little slideshow that might illustrate this."},{"Start":"08:23.795 ","End":"08:28.820","Text":"Here\u0027s something that might help and might not, give it a shot."},{"Start":"08:28.820 ","End":"08:30.920","Text":"To look at it as a game,"},{"Start":"08:30.920 ","End":"08:32.810","Text":"let\u0027s say that the error,"},{"Start":"08:32.810 ","End":"08:35.480","Text":"epsilon, and the tolerance,"},{"Start":"08:35.480 ","End":"08:38.300","Text":"delta, are involved in a game."},{"Start":"08:38.300 ","End":"08:47.420","Text":"We have to decide if we have the limit as x goes to a of f of x and I or you,"},{"Start":"08:47.420 ","End":"08:52.930","Text":"the hero is player 1 and we choose L to be the limit,"},{"Start":"08:52.930 ","End":"08:54.980","Text":"and then the opponent player,"},{"Start":"08:54.980 ","End":"08:59.690","Text":"he proposes an error level around L, that\u0027s the epsilon."},{"Start":"08:59.690 ","End":"09:03.795","Text":"He gives us an epsilon and given that epsilon,"},{"Start":"09:03.795 ","End":"09:06.870","Text":"we choose a delta,"},{"Start":"09:06.870 ","End":"09:13.619","Text":"we try to so that all the x points within a tolerance of delta"},{"Start":"09:13.619 ","End":"09:21.935","Text":"from around a produce y-values with the help of f within the error level."},{"Start":"09:21.935 ","End":"09:26.515","Text":"If we, player 1, can always win,"},{"Start":"09:26.515 ","End":"09:29.015","Text":"then that means that this is the limit,"},{"Start":"09:29.015 ","End":"09:32.010","Text":"but if even 1 time our opponent, player 2,"},{"Start":"09:32.010 ","End":"09:34.950","Text":"give us an epsilon and we can\u0027t find a delta for it,"},{"Start":"09:34.950 ","End":"09:43.480","Text":"then we\u0027ve lost and we can\u0027t say that this limit is L. Here\u0027s our function f in blue."},{"Start":"09:43.480 ","End":"09:47.605","Text":"We have a here and we claim that the limit"},{"Start":"09:47.605 ","End":"09:52.915","Text":"is L. Then the other player gives us an epsilon,"},{"Start":"09:52.915 ","End":"09:55.300","Text":"this is L plus or minus epsilon,"},{"Start":"09:55.300 ","End":"09:59.330","Text":"so we have horizontal strip or band,"},{"Start":"09:59.330 ","End":"10:04.710","Text":"and then it\u0027s our turn and we try this delta around a,"},{"Start":"10:04.710 ","End":"10:09.335","Text":"but this delta we found is not good, it\u0027s too big."},{"Start":"10:09.335 ","End":"10:14.645","Text":"We call it the tolerance in this game because if we look at it,"},{"Start":"10:14.645 ","End":"10:21.650","Text":"this part of the function is outside of the horizontal strip."},{"Start":"10:21.650 ","End":"10:23.735","Text":"That\u0027s not good, but we say, wait a minute,"},{"Start":"10:23.735 ","End":"10:25.385","Text":"I want to try again,"},{"Start":"10:25.385 ","End":"10:27.580","Text":"let\u0027s try a smaller delta."},{"Start":"10:27.580 ","End":"10:30.510","Text":"So we shrink it a bit, but oops,"},{"Start":"10:30.510 ","End":"10:36.210","Text":"it\u0027s still not good, still too big."},{"Start":"10:36.210 ","End":"10:41.265","Text":"We try again and this time, yeah,"},{"Start":"10:41.265 ","End":"10:51.239","Text":"everything\u0027s just fine because this whole thing is completely inside the horizontal band."},{"Start":"10:51.350 ","End":"10:55.660","Text":"Very good. This delta,"},{"Start":"10:55.880 ","End":"11:04.600","Text":"a plus or minus delta is good for the epsilon that was given, and that\u0027s fine."},{"Start":"11:04.790 ","End":"11:11.950","Text":"Looks good. Now, notice that if I take an even smaller delta,"},{"Start":"11:11.950 ","End":"11:13.360","Text":"that will also work."},{"Start":"11:13.360 ","End":"11:15.130","Text":"This is true in general."},{"Start":"11:15.130 ","End":"11:19.445","Text":"So that as soon as we have 1 tolerance or delta,"},{"Start":"11:19.445 ","End":"11:24.595","Text":"we can always shrink it and it will certainly still fit in."},{"Start":"11:24.595 ","End":"11:26.020","Text":"That\u0027s something to bear in mind."},{"Start":"11:26.020 ","End":"11:28.465","Text":"Delta is not unique and when we have a delta,"},{"Start":"11:28.465 ","End":"11:31.775","Text":"a tolerance, anything smaller will also work."},{"Start":"11:31.775 ","End":"11:36.790","Text":"So we won that round and now the opponent tries a different epsilon,"},{"Start":"11:36.790 ","End":"11:42.980","Text":"a smaller, a narrower band and challenges us to find a delta."},{"Start":"11:42.980 ","End":"11:47.740","Text":"This is our reply that we can take this tolerance."},{"Start":"11:47.740 ","End":"11:53.225","Text":"The delta is pretty small here and that seems to fit in."},{"Start":"11:53.225 ","End":"11:55.520","Text":"The part of the function that\u0027s within"},{"Start":"11:55.520 ","End":"12:02.430","Text":"the vertical strip is also completely inside the horizontal strip."},{"Start":"12:02.930 ","End":"12:06.720","Text":"We win this round too and so on and so on,"},{"Start":"12:06.720 ","End":"12:10.430","Text":"and I hope this was useful to you."},{"Start":"12:10.430 ","End":"12:15.365","Text":"Let\u0027s get back to where we were and now it\u0027s time for an example."},{"Start":"12:15.365 ","End":"12:19.290","Text":"For the example, I would like to clear a bit of space."},{"Start":"12:19.290 ","End":"12:21.990","Text":"I\u0027ll remove the informal stuff,"},{"Start":"12:21.990 ","End":"12:23.630","Text":"move the definition over here,"},{"Start":"12:23.630 ","End":"12:25.430","Text":"and then I can put the picture up here,"},{"Start":"12:25.430 ","End":"12:28.650","Text":"and I think we will scroll back to the top."}],"ID":8305},{"Watched":false,"Name":"Definition of Limit - Examples","Duration":"13m 25s","ChapterTopicVideoID":8156,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:11.580","Text":"The example I want to do is the limit as x goes to 2 of 3x minus 1,"},{"Start":"00:11.580 ","End":"00:12.990","Text":"I\u0027ll put that in brackets,"},{"Start":"00:12.990 ","End":"00:15.750","Text":"is equal to 5."},{"Start":"00:15.750 ","End":"00:18.885","Text":"I want to show this using Epsilon-Delta."},{"Start":"00:18.885 ","End":"00:22.500","Text":"We\u0027re going to use a definition which means that if someone gives"},{"Start":"00:22.500 ","End":"00:26.580","Text":"us any number Epsilon, which is bigger than 0,"},{"Start":"00:26.580 ","End":"00:35.055","Text":"we have to provide a Delta such that this holds whenever this holds."},{"Start":"00:35.055 ","End":"00:44.030","Text":"It means that 0 less than x minus a less than Delta."},{"Start":"00:44.030 ","End":"00:49.985","Text":"Then logical implication is that absolute value,"},{"Start":"00:49.985 ","End":"00:55.355","Text":"f of x minus L is less than Epsilon."},{"Start":"00:55.355 ","End":"00:59.210","Text":"We\u0027re given Epsilon and we have to find Delta."},{"Start":"00:59.210 ","End":"01:01.310","Text":"Now, since this is our first example,"},{"Start":"01:01.310 ","End":"01:03.274","Text":"I\u0027m going to go extra slow."},{"Start":"01:03.274 ","End":"01:06.260","Text":"I\u0027m going to do something which is not done in most books,"},{"Start":"01:06.260 ","End":"01:10.039","Text":"is to actually start off with a specific epsilon example."},{"Start":"01:10.039 ","End":"01:17.255","Text":"Suppose I give you epsilon equals 0.015,"},{"Start":"01:17.255 ","End":"01:22.965","Text":"then we have to find a matching Delta such that all this holds."},{"Start":"01:22.965 ","End":"01:26.465","Text":"When I\u0027ve done this, then we\u0027ll go to a more general epsilon."},{"Start":"01:26.465 ","End":"01:28.760","Text":"Note that in our particular example,"},{"Start":"01:28.760 ","End":"01:38.210","Text":"a is the 2 and this 5 is the big L. F of x is this function,"},{"Start":"01:38.210 ","End":"01:43.245","Text":"it\u0027s 3x minus 1."},{"Start":"01:43.245 ","End":"01:48.005","Text":"We have to find a Delta such that if"},{"Start":"01:48.005 ","End":"01:54.920","Text":"the absolute value of x minus 2 is less than Delta and is bigger than 0,"},{"Start":"01:54.920 ","End":"01:56.435","Text":"x con, equal 2."},{"Start":"01:56.435 ","End":"01:59.250","Text":"Then if I compute this,"},{"Start":"01:59.250 ","End":"02:02.065","Text":"this is 3x minus 1 minus 5."},{"Start":"02:02.065 ","End":"02:06.350","Text":"Then we want the absolute value of 3x minus"},{"Start":"02:06.350 ","End":"02:12.560","Text":"6 to be less than epsilon, that\u0027s in general."},{"Start":"02:12.560 ","End":"02:14.900","Text":"I\u0027ll return to this in a moment,"},{"Start":"02:14.900 ","End":"02:18.250","Text":"I just want to go aside and do it for a particular epsilon."},{"Start":"02:18.250 ","End":"02:20.350","Text":"I\u0027ll copy this over here,"},{"Start":"02:20.350 ","End":"02:22.280","Text":"except instead of the Epsilon,"},{"Start":"02:22.280 ","End":"02:26.330","Text":"I\u0027m going to put 0.015."},{"Start":"02:26.330 ","End":"02:28.940","Text":"Like I said, there\u0027s something I\u0027m doing just because it\u0027s the first time,"},{"Start":"02:28.940 ","End":"02:31.620","Text":"normally you wouldn\u0027t do a specific example,"},{"Start":"02:31.620 ","End":"02:34.475","Text":"you work straight away on the general Epsilon."},{"Start":"02:34.475 ","End":"02:39.130","Text":"Now what I\u0027m going to do is just work on this inequality."},{"Start":"02:39.130 ","End":"02:45.465","Text":"I notice that I can take 3 outside the brackets of 3x minus 6."},{"Start":"02:45.465 ","End":"02:53.625","Text":"I\u0027ve got the absolute value of 3 times x minus 2 less than 0.015."},{"Start":"02:53.625 ","End":"03:01.950","Text":"That gives me 3 times the absolute value of x minus 2 is less than 0.015."},{"Start":"03:03.020 ","End":"03:05.980","Text":"Because 3 is positive,"},{"Start":"03:05.980 ","End":"03:10.615","Text":"we can divide both sides of the inequality by it and so we get"},{"Start":"03:10.615 ","End":"03:15.870","Text":"x minus 2 less than this over 3, this is what we get."},{"Start":"03:15.870 ","End":"03:20.800","Text":"This looks very much like this or at least this part so I"},{"Start":"03:20.800 ","End":"03:30.085","Text":"suggest that we take our Delta to equal 0.005."},{"Start":"03:30.085 ","End":"03:36.640","Text":"Now we have to do the work again because I want to start with"},{"Start":"03:36.640 ","End":"03:43.750","Text":"this assumption and then through a series of steps to reach this at the end."},{"Start":"03:43.750 ","End":"03:48.050","Text":"What we can start with is using this Delta,"},{"Start":"03:48.050 ","End":"03:57.350","Text":"we have that 0 less than absolute value of x minus 2 less than 0.005."},{"Start":"03:57.350 ","End":"04:02.030","Text":"I can actually ignore the 0 because if this is true,"},{"Start":"04:02.030 ","End":"04:08.905","Text":"then certainly absolute value of x minus 2 is less than 0.005."},{"Start":"04:08.905 ","End":"04:12.210","Text":"Now multiply both sides by 3."},{"Start":"04:12.210 ","End":"04:18.530","Text":"3 times absolute value of x minus 2 is less than 0.0153,"},{"Start":"04:18.530 ","End":"04:21.840","Text":"3 is positive so I can do that."},{"Start":"04:21.890 ","End":"04:25.260","Text":"Then if I put the 3 inside,"},{"Start":"04:25.260 ","End":"04:32.560","Text":"I\u0027ve got that 3x minus 6 less than 0.015."},{"Start":"04:33.200 ","End":"04:37.545","Text":"This is what we wanted to get to so we\u0027re okay."},{"Start":"04:37.545 ","End":"04:40.395","Text":"We\u0027ve done Epsilon equals 0.015."},{"Start":"04:40.395 ","End":"04:43.525","Text":"Now let\u0027s do it for a general Epsilon."},{"Start":"04:43.525 ","End":"04:45.875","Text":"In the general case of epsilon,"},{"Start":"04:45.875 ","End":"04:48.905","Text":"or just have to mimic what I did in this particular case."},{"Start":"04:48.905 ","End":"04:51.050","Text":"I could take the 3 out,"},{"Start":"04:51.050 ","End":"04:54.470","Text":"3 is positive so I can take it out of the absolute value,"},{"Start":"04:54.470 ","End":"04:58.310","Text":"3 times x minus 2 less than epsilon,"},{"Start":"04:58.310 ","End":"04:59.510","Text":"then divide by 3."},{"Start":"04:59.510 ","End":"05:04.835","Text":"It\u0027s a positive quantity so I can do that epsilon over 3."},{"Start":"05:04.835 ","End":"05:15.050","Text":"Then it looks like I could take my Delta equals Epsilon over 3."},{"Start":"05:15.050 ","End":"05:17.540","Text":"If I take Delta equals Epsilon over 3,"},{"Start":"05:17.540 ","End":"05:22.080","Text":"now I have to show again that this becomes true."},{"Start":"05:22.080 ","End":"05:23.670","Text":"I don\u0027t have to use this part,"},{"Start":"05:23.670 ","End":"05:27.450","Text":"I can just use this part of the inequality so that absolute value of"},{"Start":"05:27.450 ","End":"05:31.785","Text":"x minus 2 less than epsilon over 3."},{"Start":"05:31.785 ","End":"05:33.930","Text":"Multiply both sides by 3,"},{"Start":"05:33.930 ","End":"05:39.705","Text":"and I\u0027ve got 3 absolute value of x minus 2 less than epsilon."},{"Start":"05:39.705 ","End":"05:45.530","Text":"Then I put the 3 inside and get that absolute value of"},{"Start":"05:45.530 ","End":"05:51.765","Text":"3x minus 6 is less than epsilon."},{"Start":"05:51.765 ","End":"05:54.635","Text":"This is just exactly what we had to show,"},{"Start":"05:54.635 ","End":"05:58.050","Text":"only we had to start from here and up here."},{"Start":"05:58.700 ","End":"06:02.825","Text":"We\u0027ve basically done this example."},{"Start":"06:02.825 ","End":"06:09.880","Text":"We showed that for each Epsilon we can actually take Delta is equal to epsilon over 3,"},{"Start":"06:09.880 ","End":"06:12.380","Text":"could take smaller also would also work,"},{"Start":"06:12.380 ","End":"06:15.575","Text":"but this is one possibility and this works."},{"Start":"06:15.575 ","End":"06:20.585","Text":"I would just like to cover a small technical but important point."},{"Start":"06:20.585 ","End":"06:23.510","Text":"Suppose I had a slightly different example."},{"Start":"06:23.510 ","End":"06:29.860","Text":"Suppose it wasn\u0027t written this way but I had that f of x,"},{"Start":"06:29.860 ","End":"06:37.290","Text":"instead of being 3x minus 1 suppose I took f of x to be 3x minus"},{"Start":"06:37.290 ","End":"06:44.745","Text":"1 when x is not equal to 2."},{"Start":"06:44.745 ","End":"06:50.030","Text":"Suppose I took it to be 6 when x equals 2,"},{"Start":"06:50.030 ","End":"06:51.650","Text":"in other words, f of 2 is 6."},{"Start":"06:51.650 ","End":"06:58.525","Text":"What I\u0027m saying is if this was our picture and this was 2 and this was 5,"},{"Start":"06:58.525 ","End":"07:03.245","Text":"I\u0027m going to change the function to make this point be up here,"},{"Start":"07:03.245 ","End":"07:06.020","Text":"so the 5 to be at 6."},{"Start":"07:06.020 ","End":"07:08.045","Text":"The question is then,"},{"Start":"07:08.045 ","End":"07:15.000","Text":"what would be the limit as x goes to 2 of f of x?"},{"Start":"07:15.610 ","End":"07:18.115","Text":"If you think about it,"},{"Start":"07:18.115 ","End":"07:20.665","Text":"the answer is that nothing changes."},{"Start":"07:20.665 ","End":"07:25.765","Text":"Even if I change the value at the point x equals 2 to 6 or to whatever,"},{"Start":"07:25.765 ","End":"07:29.830","Text":"or even if I just throw it away and don\u0027t define it there,"},{"Start":"07:29.830 ","End":"07:37.040","Text":"the limit will not change because only the values that are close to 2 matter."},{"Start":"07:37.200 ","End":"07:40.060","Text":"If we wanted to do it, technically,"},{"Start":"07:40.060 ","End":"07:44.610","Text":"you would see that when I plugged in f of x here,"},{"Start":"07:44.610 ","End":"07:46.215","Text":"it was 3x minus 1,"},{"Start":"07:46.215 ","End":"07:51.340","Text":"that was okay because here x is not equal to 2."},{"Start":"07:51.340 ","End":"07:53.770","Text":"Because of this bigger than 0,"},{"Start":"07:53.770 ","End":"07:57.100","Text":"x is not equal to 2 and therefore,"},{"Start":"07:57.100 ","End":"08:03.785","Text":"I could know that its value is from the top row here."},{"Start":"08:03.785 ","End":"08:08.990","Text":"It just bothered me that we didn\u0027t use this fact bigger than 0 here and it"},{"Start":"08:08.990 ","End":"08:14.370","Text":"is important because this\u0027s what keeps the x away from being 2 here."},{"Start":"08:16.040 ","End":"08:21.635","Text":"I wanted to show you that if you actually change the value at the given point,"},{"Start":"08:21.635 ","End":"08:24.605","Text":"it doesn\u0027t affect the limit, I keep saying it."},{"Start":"08:24.605 ","End":"08:28.700","Text":"The limit only cares about what happens near the point a,"},{"Start":"08:28.700 ","End":"08:31.880","Text":"doesn\u0027t care about what actually happens at the point a,"},{"Start":"08:31.880 ","End":"08:35.450","Text":"what the function or achieving if it\u0027s defined there."},{"Start":"08:35.450 ","End":"08:37.880","Text":"Anyway, we\u0027re done with this example."},{"Start":"08:37.880 ","End":"08:41.330","Text":"Now, I\u0027ll do one more example."},{"Start":"08:41.330 ","End":"08:45.670","Text":"We\u0027ll just do it more quickly because it\u0027s a second example."},{"Start":"08:45.670 ","End":"08:49.655","Text":"This example will be the following limit."},{"Start":"08:49.655 ","End":"08:55.535","Text":"Limit as x goes to 0 of x squared equals 0."},{"Start":"08:55.535 ","End":"08:59.915","Text":"Looks obvious, but we have to do it with Epsilon-Delta."},{"Start":"08:59.915 ","End":"09:03.680","Text":"What we have to show is for every number epsilon,"},{"Start":"09:03.680 ","End":"09:11.150","Text":"that\u0027s the one we\u0027re given so let\u0027s say someone gives us an Epsilon bigger than 0,"},{"Start":"09:11.150 ","End":"09:17.385","Text":"we need to find another number Delta bigger than 0,"},{"Start":"09:17.385 ","End":"09:22.775","Text":"such that, such that I write as st."},{"Start":"09:22.775 ","End":"09:24.800","Text":"Instead of saying this,"},{"Start":"09:24.800 ","End":"09:27.500","Text":"whenever this, I\u0027ll write that this implies this,"},{"Start":"09:27.500 ","End":"09:36.590","Text":"such that 0 less than absolute value of x minus a less than"},{"Start":"09:36.590 ","End":"09:46.910","Text":"Delta implies that absolute value of f of x minus L less than Epsilon."},{"Start":"09:46.910 ","End":"09:50.660","Text":"But I want to write the quantities that we know."},{"Start":"09:50.660 ","End":"09:55.440","Text":"We know that a is 0 because that\u0027s this,"},{"Start":"09:55.440 ","End":"09:57.780","Text":"we know that L is 0, that\u0027s this."},{"Start":"09:57.780 ","End":"10:00.820","Text":"We know f of x is x squared."},{"Start":"10:01.190 ","End":"10:03.780","Text":"I\u0027ll just write the quantities in,"},{"Start":"10:03.780 ","End":"10:06.245","Text":"x minus 0 less than Delta,"},{"Start":"10:06.245 ","End":"10:10.480","Text":"but x is not 0, implies that x squared,"},{"Start":"10:10.480 ","End":"10:15.930","Text":"which is our f of x minus 0,"},{"Start":"10:15.930 ","End":"10:19.405","Text":"that\u0027s going to be less than Epsilon."},{"Start":"10:19.405 ","End":"10:21.195","Text":"Let\u0027s slightly rewrite it,"},{"Start":"10:21.195 ","End":"10:22.665","Text":"there\u0027s too many 0s here."},{"Start":"10:22.665 ","End":"10:26.530","Text":"What we have to find the Delta such that whenever"},{"Start":"10:26.530 ","End":"10:32.845","Text":"absolute value of x is less than delta and also x is not 0,"},{"Start":"10:32.845 ","End":"10:39.460","Text":"then we have that the absolute value of x squared is less than Epsilon."},{"Start":"10:39.460 ","End":"10:41.140","Text":"We\u0027re given Epsilon,"},{"Start":"10:41.140 ","End":"10:43.205","Text":"we have to find Delta."},{"Start":"10:43.205 ","End":"10:48.890","Text":"What we do is we usually start with this condition on"},{"Start":"10:48.890 ","End":"10:51.440","Text":"the Epsilon and see if we can"},{"Start":"10:51.440 ","End":"10:55.370","Text":"manipulate it algebraically and that will give us a hint as to what Delta is."},{"Start":"10:55.370 ","End":"11:00.840","Text":"Then we have to actually do second time the verification phase."},{"Start":"11:00.840 ","End":"11:02.610","Text":"Start with this,"},{"Start":"11:02.610 ","End":"11:05.325","Text":"x squared less than Epsilon."},{"Start":"11:05.325 ","End":"11:07.340","Text":"If we take the square root of that,"},{"Start":"11:07.340 ","End":"11:12.145","Text":"we\u0027ve got absolute value of x less than square root of Epsilon."},{"Start":"11:12.145 ","End":"11:14.780","Text":"If you\u0027re not sure of that,"},{"Start":"11:14.780 ","End":"11:19.315","Text":"just think if I said x squared is less than 9,"},{"Start":"11:19.315 ","End":"11:24.365","Text":"wouldn\u0027t that imply that absolute value of x is less than 3?"},{"Start":"11:24.365 ","End":"11:28.325","Text":"It seems to suggest if looking at this,"},{"Start":"11:28.325 ","End":"11:32.085","Text":"that we should take as Delta,"},{"Start":"11:32.085 ","End":"11:35.565","Text":"at least try Delta equals square root of Epsilon."},{"Start":"11:35.565 ","End":"11:42.455","Text":"I\u0027m going to say let\u0027s try Delta equals the square root of Epsilon."},{"Start":"11:42.455 ","End":"11:45.575","Text":"Now we have to do it in the forward direction."},{"Start":"11:45.575 ","End":"11:49.520","Text":"Start from here and end up here."},{"Start":"11:49.520 ","End":"11:58.500","Text":"Well, if absolute value of x is between 0 and Delta,"},{"Start":"11:58.500 ","End":"12:01.625","Text":"these are 2 inequalities and I can just ignore this."},{"Start":"12:01.625 ","End":"12:06.980","Text":"In particular, this is true and this I can ignore."},{"Start":"12:06.980 ","End":"12:09.995","Text":"If absolute value of x is less than Delta,"},{"Start":"12:09.995 ","End":"12:14.630","Text":"we are allowed to take the square of both sides."},{"Start":"12:14.630 ","End":"12:21.515","Text":"Absolute value of x squared is less than Delta squared."},{"Start":"12:21.515 ","End":"12:25.850","Text":"Now, absolute value of x squared is the same as the absolute value of x"},{"Start":"12:25.850 ","End":"12:30.810","Text":"squared and Delta squared is equal to Epsilon."},{"Start":"12:31.100 ","End":"12:35.750","Text":"Just replacing by what they\u0027re equal to, the inequality stays."},{"Start":"12:35.750 ","End":"12:37.820","Text":"This is what we had to show."},{"Start":"12:37.820 ","End":"12:43.195","Text":"We started from this and ended up in this so we\u0027re fine."},{"Start":"12:43.195 ","End":"12:49.265","Text":"That\u0027s another example of a limit."},{"Start":"12:49.265 ","End":"12:55.250","Text":"Again, I wanted to just say by the way that if I chose a function which"},{"Start":"12:55.250 ","End":"13:00.710","Text":"was x squared everywhere except 0 and redefined it at 0 or even undefined it at 0,"},{"Start":"13:00.710 ","End":"13:07.330","Text":"it wouldn\u0027t matter because this guarantees that my x is not 0."},{"Start":"13:07.330 ","End":"13:10.130","Text":"If I change the value of the function at 0,"},{"Start":"13:10.130 ","End":"13:15.290","Text":"it\u0027s not going to affect this x which is not 0."},{"Start":"13:15.290 ","End":"13:22.830","Text":"Anyway, that\u0027s it. I think that\u0027ll do for the examples for this part."}],"ID":8310},{"Watched":false,"Name":"One-Sided Limit","Duration":"8m 45s","ChapterTopicVideoID":8152,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.770","Text":"In the previous clip,"},{"Start":"00:01.770 ","End":"00:10.605","Text":"we talked about the definition of a limit of a function using Epsilon and Delta."},{"Start":"00:10.605 ","End":"00:16.725","Text":"Here, we\u0027re going to talk about something called a one-sided limit."},{"Start":"00:16.725 ","End":"00:18.480","Text":"In fact, there\u0027ll be 2 of them,"},{"Start":"00:18.480 ","End":"00:20.895","Text":"there\u0027ll be a right limit and a left limit."},{"Start":"00:20.895 ","End":"00:24.375","Text":"Let me just go back to the regular limit."},{"Start":"00:24.375 ","End":"00:30.665","Text":"We said that limit x goes to a"},{"Start":"00:30.665 ","End":"00:38.735","Text":"of f of x is equal to L if certain conditions held."},{"Start":"00:38.735 ","End":"00:45.680","Text":"Well, f had to be defined in a certain interval around x equals a,"},{"Start":"00:45.680 ","End":"00:49.260","Text":"but not necessarily at x equals a."},{"Start":"00:49.260 ","End":"00:52.035","Text":"Then we had an Epsilon-Delta,"},{"Start":"00:52.035 ","End":"00:57.730","Text":"that if Epsilon is bigger than 0,"},{"Start":"00:57.730 ","End":"01:02.330","Text":"then we can find Delta bigger than 0 such that"},{"Start":"01:02.330 ","End":"01:09.000","Text":"if x minus a is less than Delta,"},{"Start":"01:09.000 ","End":"01:12.425","Text":"actually bigger than 0,"},{"Start":"01:12.425 ","End":"01:17.790","Text":"then we have that f of x minus L."},{"Start":"01:17.790 ","End":"01:22.550","Text":"Absolute value is less than Epsilon."},{"Start":"01:22.550 ","End":"01:25.939","Text":"I\u0027m just summarizing it, didn\u0027t write all the words."},{"Start":"01:25.939 ","End":"01:32.800","Text":"We\u0027re going to modify this for the one-sided limits."},{"Start":"01:32.800 ","End":"01:36.740","Text":"This, by the way, is called a two-sided limit also."},{"Start":"01:36.740 ","End":"01:40.910","Text":"Now, the one-sided limit from the right will be written"},{"Start":"01:40.910 ","End":"01:46.055","Text":"as limit as x goes to a, with a plus here,"},{"Start":"01:46.055 ","End":"01:48.200","Text":"say from above or from the right,"},{"Start":"01:48.200 ","End":"01:49.520","Text":"then both will work,"},{"Start":"01:49.520 ","End":"01:54.240","Text":"of f of x is equal to L."},{"Start":"01:54.240 ","End":"01:56.590","Text":"For one thing,"},{"Start":"01:59.270 ","End":"02:03.155","Text":"I can\u0027t say some interval containing a,"},{"Start":"02:03.155 ","End":"02:08.210","Text":"but f has to be defined on"},{"Start":"02:08.210 ","End":"02:12.770","Text":"some interval to the right of a from a to something,"},{"Start":"02:12.770 ","End":"02:16.160","Text":"that\u0027s a to b or could be infinity."},{"Start":"02:16.160 ","End":"02:19.160","Text":"Don\u0027t worry too much about the definitions and domain."},{"Start":"02:19.160 ","End":"02:20.795","Text":"Not that important."},{"Start":"02:20.795 ","End":"02:25.730","Text":"The right limit says that for every Epsilon, there is a Delta,"},{"Start":"02:25.730 ","End":"02:29.750","Text":"such that instead of this,"},{"Start":"02:29.750 ","End":"02:35.340","Text":"we\u0027re going to have just without the absolute value."},{"Start":"02:38.150 ","End":"02:40.220","Text":"If this is true,"},{"Start":"02:40.220 ","End":"02:43.040","Text":"then the same thing here holds true."},{"Start":"02:43.040 ","End":"02:45.470","Text":"f of x minus L,"},{"Start":"02:45.470 ","End":"02:48.410","Text":"I\u0027ll erase this,"},{"Start":"02:48.410 ","End":"02:51.305","Text":"less than Epsilon."},{"Start":"02:51.305 ","End":"02:58.490","Text":"Perhaps, I will write that for each Epsilon bigger than 0,"},{"Start":"02:58.490 ","End":"03:04.085","Text":"there is Delta bigger than 0 such that,"},{"Start":"03:04.085 ","End":"03:08.240","Text":"that holds for this and for this and for the left limit."},{"Start":"03:08.240 ","End":"03:10.010","Text":"This is rather than brackets."},{"Start":"03:10.010 ","End":"03:12.620","Text":"That\u0027s the right limit."},{"Start":"03:12.620 ","End":"03:16.610","Text":"Now, this is the two-sided limit or just the plain limit."},{"Start":"03:16.610 ","End":"03:23.180","Text":"The left limit will be x goes to a and a minus here."},{"Start":"03:23.180 ","End":"03:24.920","Text":"Just a symbol."},{"Start":"03:24.920 ","End":"03:27.710","Text":"This means limit from the right or limit from above."},{"Start":"03:27.710 ","End":"03:31.020","Text":"Limit from the left, limit from below."},{"Start":"03:31.120 ","End":"03:33.740","Text":"f of x is L."},{"Start":"03:33.740 ","End":"03:36.740","Text":"If for each Epsilon, there is a Delta,"},{"Start":"03:36.740 ","End":"03:38.704","Text":"such that when"},{"Start":"03:38.704 ","End":"03:46.385","Text":"minus Delta x minus a less than 0 implies the same thing,"},{"Start":"03:46.385 ","End":"03:54.490","Text":"they will end the same way that f of x is close to L within an era of Epsilon."},{"Start":"03:54.490 ","End":"03:59.975","Text":"These 2 are sometimes slightly rewritten."},{"Start":"03:59.975 ","End":"04:02.059","Text":"Just like here."},{"Start":"04:02.059 ","End":"04:08.660","Text":"I could write this without absolute value and say that x is between"},{"Start":"04:08.660 ","End":"04:17.010","Text":"a plus Delta and a minus Delta,"},{"Start":"04:17.010 ","End":"04:22.490","Text":"but with the proviso that x is not equal to a."},{"Start":"04:22.490 ","End":"04:24.245","Text":"That\u0027s what this translates to."},{"Start":"04:24.245 ","End":"04:29.270","Text":"This just translates to if you just want to condition on x,"},{"Start":"04:29.270 ","End":"04:35.100","Text":"that x is between a plus Delta and a."},{"Start":"04:35.100 ","End":"04:37.159","Text":"This is often more convenient."},{"Start":"04:37.159 ","End":"04:41.115","Text":"Obviously, we just add a to all 3 sides of this inequality."},{"Start":"04:41.115 ","End":"04:51.690","Text":"Here we would get that a minus Delta less than x, less than a."},{"Start":"04:52.370 ","End":"04:58.355","Text":"This one is the left limit or the limit from below."},{"Start":"04:58.355 ","End":"05:02.810","Text":"I\u0027ll just give an example of one of them."},{"Start":"05:02.810 ","End":"05:06.755","Text":"I\u0027ll give an example of the right limit."},{"Start":"05:06.755 ","End":"05:14.480","Text":"The example will be to prove that the limit as"},{"Start":"05:14.480 ","End":"05:22.775","Text":"x goes to 0 from the right of the square root of x is equal to 0."},{"Start":"05:22.775 ","End":"05:25.775","Text":"Just noting that this function square root of x,"},{"Start":"05:25.775 ","End":"05:27.230","Text":"we think of it as f of x"},{"Start":"05:27.230 ","End":"05:32.520","Text":"is defined on the interval from 0 to infinity."},{"Start":"05:33.880 ","End":"05:36.635","Text":"x bigger than 0."},{"Start":"05:36.635 ","End":"05:40.705","Text":"But it\u0027s defined in any event close to 0,"},{"Start":"05:40.705 ","End":"05:44.210","Text":"as close as we want to 0 and on the right of 0."},{"Start":"05:44.210 ","End":"05:47.329","Text":"It\u0027s actually defined at x bigger or equal to 0,"},{"Start":"05:47.329 ","End":"05:50.865","Text":"but I don\u0027t need the equals 0."},{"Start":"05:50.865 ","End":"05:52.340","Text":"How do we prove this?"},{"Start":"05:52.340 ","End":"05:53.719","Text":"The Epsilon-Delta."},{"Start":"05:53.719 ","End":"05:58.880","Text":"Someone gives us Epsilon bigger than 0 is given."},{"Start":"05:58.880 ","End":"06:01.325","Text":"We have to find the Delta."},{"Start":"06:01.325 ","End":"06:05.645","Text":"Let me just indicate Epsilon bigger than 0 is given."},{"Start":"06:05.645 ","End":"06:13.980","Text":"We need to find the Delta for this Epsilon."},{"Start":"06:13.980 ","End":"06:24.090","Text":"What does this Delta had to satisfy if 0 is less than x?"},{"Start":"06:24.090 ","End":"06:28.650","Text":"Now notice that a is 0. Please make a note of that."},{"Start":"06:28.650 ","End":"06:30.830","Text":"In our case, a is 0,"},{"Start":"06:30.830 ","End":"06:35.420","Text":"L is 0, and f of x is the square root of x."},{"Start":"06:35.420 ","End":"06:37.370","Text":"If I apply all that,"},{"Start":"06:37.370 ","End":"06:44.690","Text":"then we have to find Delta such that 0 is less than x."},{"Start":"06:44.690 ","End":"06:46.355","Text":"x minus 0."},{"Start":"06:46.355 ","End":"06:50.840","Text":"Less than Delta implies"},{"Start":"06:50.840 ","End":"06:57.965","Text":"that the absolute value of the square root of x minus 0,"},{"Start":"06:57.965 ","End":"07:00.680","Text":"square root of x minus 0 is positive,"},{"Start":"07:00.680 ","End":"07:03.110","Text":"so I don\u0027t need the absolute value."},{"Start":"07:03.110 ","End":"07:08.230","Text":"Square root of x less than Epsilon."},{"Start":"07:08.960 ","End":"07:12.080","Text":"How do we find such a Delta?"},{"Start":"07:12.080 ","End":"07:17.195","Text":"As before, you start from the end from this and say, well,"},{"Start":"07:17.195 ","End":"07:20.330","Text":"if square root of x is less than Epsilon,"},{"Start":"07:20.330 ","End":"07:25.730","Text":"then we could say that if you square both sides,"},{"Start":"07:25.730 ","End":"07:29.730","Text":"that x is less than Epsilon squared."},{"Start":"07:30.740 ","End":"07:40.030","Text":"What we would do would be to try Delta is equal to Epsilon squared and see if that works."},{"Start":"07:40.030 ","End":"07:41.560","Text":"Then we start again."},{"Start":"07:41.560 ","End":"07:43.405","Text":"If this is true,"},{"Start":"07:43.405 ","End":"07:48.459","Text":"then assuming that Delta equals Epsilon squared,"},{"Start":"07:48.459 ","End":"07:52.750","Text":"then we have that 0 is less than x,"},{"Start":"07:52.750 ","End":"07:55.990","Text":"less than Epsilon squared,"},{"Start":"07:55.990 ","End":"07:59.710","Text":"and then x is positive and less than Epsilon squared."},{"Start":"07:59.710 ","End":"08:03.970","Text":"We can take the square root of everything."},{"Start":"08:03.970 ","End":"08:08.740","Text":"This would imply that square root of 0"},{"Start":"08:08.740 ","End":"08:15.735","Text":"less than square root of x less than Epsilon."},{"Start":"08:15.735 ","End":"08:22.120","Text":"In particular, I can ignore the fact that 0 less than,"},{"Start":"08:22.120 ","End":"08:23.770","Text":"I just need this bit."},{"Start":"08:23.770 ","End":"08:26.540","Text":"This gives me this, I mean,"},{"Start":"08:26.540 ","End":"08:30.110","Text":"it is that we proved it for each Epsilon,"},{"Start":"08:30.110 ","End":"08:32.450","Text":"we take Delta equals Epsilon squared."},{"Start":"08:32.450 ","End":"08:35.330","Text":"Then whenever this is true, this is true,"},{"Start":"08:35.330 ","End":"08:38.915","Text":"which is what we have to show for the right-handed limit."},{"Start":"08:38.915 ","End":"08:41.500","Text":"I won\u0027t give an example of a left-handed limited."},{"Start":"08:41.500 ","End":"08:43.085","Text":"It\u0027s very similar."},{"Start":"08:43.085 ","End":"08:46.260","Text":"We\u0027ll just end the clip here."}],"ID":8306},{"Watched":false,"Name":"Infinite Limit","Duration":"13m 29s","ChapterTopicVideoID":8153,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"Now I want to introduce slight variation on the definition of a limit."},{"Start":"00:05.490 ","End":"00:07.710","Text":"There are things called infinite limits,"},{"Start":"00:07.710 ","End":"00:14.805","Text":"and I\u0027ll show you what this is and I\u0027d like to compare it to a regular limit."},{"Start":"00:14.805 ","End":"00:17.415","Text":"To start off with, I just copy"},{"Start":"00:17.415 ","End":"00:23.840","Text":"the case of a regular limit and I\u0027m going to show you how we change it."},{"Start":"00:23.840 ","End":"00:28.310","Text":"This was what we wanted to"},{"Start":"00:28.310 ","End":"00:30.980","Text":"define and this is how we formally"},{"Start":"00:30.980 ","End":"00:34.685","Text":"defined it and informally and there was a diagram also."},{"Start":"00:34.685 ","End":"00:39.185","Text":"Now the big change is going to be that instead of this number L,"},{"Start":"00:39.185 ","End":"00:41.545","Text":"I\u0027m going to put in infinity."},{"Start":"00:41.545 ","End":"00:46.990","Text":"Here instead of L, we have infinity and let\u0027s see what else we\u0027ll have to change."},{"Start":"00:46.990 ","End":"00:49.850","Text":"Well, for 1 thing, this diagram is no use"},{"Start":"00:49.850 ","End":"00:53.240","Text":"anymore because if L is infinity it won\u0027t look like this."},{"Start":"00:53.240 ","End":"00:55.730","Text":"I\u0027d like to start with the informal definition."},{"Start":"00:55.730 ","End":"00:59.555","Text":"We can make f of x as close as we like to instead of L,"},{"Start":"00:59.555 ","End":"01:07.070","Text":"I\u0027m going to put infinity by making x close enough to a not equal to a."},{"Start":"01:07.070 ","End":"01:09.020","Text":"Now, here\u0027s what I have to explain."},{"Start":"01:09.020 ","End":"01:11.990","Text":"What does it mean as close as we like to infinity?"},{"Start":"01:11.990 ","End":"01:14.260","Text":"How should I interpret that?"},{"Start":"01:14.260 ","End":"01:16.070","Text":"If you think about it,"},{"Start":"01:16.070 ","End":"01:20.615","Text":"being close to infinity means being very, very large."},{"Start":"01:20.615 ","End":"01:24.125","Text":"Instead of saying as close as we like to infinity,"},{"Start":"01:24.125 ","End":"01:27.275","Text":"I\u0027ll write as large as we like."},{"Start":"01:27.275 ","End":"01:34.405","Text":"Here, as large as we like."},{"Start":"01:34.405 ","End":"01:40.110","Text":"Now all I have to do is alter the formally part."},{"Start":"01:40.130 ","End":"01:44.240","Text":"When we say as close as we like to L,"},{"Start":"01:44.240 ","End":"01:48.665","Text":"we chose it to be within a certain error margin"},{"Start":"01:48.665 ","End":"01:54.590","Text":"Epsilon of L. But here we want it to be close to infinity,"},{"Start":"01:54.590 ","End":"01:58.825","Text":"so it\u0027s larger than any number we please."},{"Start":"01:58.825 ","End":"02:01.840","Text":"I\u0027m going to change the Epsilon."},{"Start":"02:01.840 ","End":"02:03.680","Text":"I could still use the same letter,"},{"Start":"02:03.680 ","End":"02:05.930","Text":"but I prefer to use a different letter."},{"Start":"02:05.930 ","End":"02:09.170","Text":"I\u0027m going to say for every number M,"},{"Start":"02:09.170 ","End":"02:11.180","Text":"so I\u0027ll say M,"},{"Start":"02:11.180 ","End":"02:12.910","Text":"for every M,"},{"Start":"02:12.910 ","End":"02:16.450","Text":"still I have the Delta."},{"Start":"02:16.450 ","End":"02:20.420","Text":"But instead of saying that f is close to L with an Epsilon,"},{"Start":"02:20.420 ","End":"02:24.600","Text":"I\u0027ll replace this whole condition by f"},{"Start":"02:24.600 ","End":"02:29.150","Text":"of x is bigger than M. That\u0027s what close to infinity is,"},{"Start":"02:29.150 ","End":"02:32.330","Text":"means larger than any number that we please."},{"Start":"02:32.330 ","End":"02:34.760","Text":"That\u0027s the analogy."},{"Start":"02:34.760 ","End":"02:37.355","Text":"This basically gives us the definition."},{"Start":"02:37.355 ","End":"02:42.300","Text":"I\u0027ll just say that although some books do write bigger than 0,"},{"Start":"02:42.300 ","End":"02:47.190","Text":"I see no reason to write that in."},{"Start":"02:48.140 ","End":"02:51.270","Text":"But a lot of books do write,"},{"Start":"02:51.270 ","End":"02:54.260","Text":"and I like this, and they say however large,"},{"Start":"02:54.260 ","End":"02:58.640","Text":"it just gives us a hint that we\u0027re going to make it very large,"},{"Start":"02:58.640 ","End":"02:59.840","Text":"as large as we like,"},{"Start":"02:59.840 ","End":"03:03.655","Text":"that would work for any M, even negative Ms."},{"Start":"03:03.655 ","End":"03:08.480","Text":"But generally M is going to be a million or a billion or a zillion or whatever,"},{"Start":"03:08.480 ","End":"03:10.880","Text":"a very large, however large."},{"Start":"03:10.880 ","End":"03:12.080","Text":"This is the definition,"},{"Start":"03:12.080 ","End":"03:14.210","Text":"for every number M however large,"},{"Start":"03:14.210 ","End":"03:16.285","Text":"there is a Delta,"},{"Start":"03:16.285 ","End":"03:19.440","Text":"depending on M. I didn\u0027t write, that is Delta of M,"},{"Start":"03:19.440 ","End":"03:25.565","Text":"such that f of x is bigger than M whenever x"},{"Start":"03:25.565 ","End":"03:29.090","Text":"minus a in absolute value is"},{"Start":"03:29.090 ","End":"03:33.680","Text":"less than Delta and is bigger than 0 because x is not going to equal a."},{"Start":"03:33.680 ","End":"03:37.465","Text":"That\u0027s the definition for infinity."},{"Start":"03:37.465 ","End":"03:40.335","Text":"Here\u0027s the diagram for this."},{"Start":"03:40.335 ","End":"03:42.390","Text":"Let me just suggest it here."},{"Start":"03:42.390 ","End":"03:46.540","Text":"Now as you see, here\u0027s the M that is given to us,"},{"Start":"03:46.540 ","End":"03:49.595","Text":"but it could be arbitrarily large."},{"Start":"03:49.595 ","End":"03:54.895","Text":"Just we\u0027ll take 1 example where M is here and then we have to find"},{"Start":"03:54.895 ","End":"04:02.830","Text":"a vertical strip or a Delta tolerance such that whenever our x is within this tolerance,"},{"Start":"04:02.830 ","End":"04:08.230","Text":"then the graph is within this strip as it is here."},{"Start":"04:08.230 ","End":"04:11.320","Text":"Just to highlight this bit,"},{"Start":"04:11.320 ","End":"04:15.640","Text":"the function seems to have a discontinuity here,"},{"Start":"04:15.640 ","End":"04:17.810","Text":"it\u0027s in 2 pieces."},{"Start":"04:18.230 ","End":"04:22.680","Text":"Yes, the highlighting is not greatest."},{"Start":"04:22.680 ","End":"04:25.605","Text":"I should you use a different color perhaps."},{"Start":"04:25.605 ","End":"04:27.445","Text":"You can see it though."},{"Start":"04:27.445 ","End":"04:31.450","Text":"This doesn\u0027t have to be exactly the intersection point."},{"Start":"04:31.450 ","End":"04:36.880","Text":"In general, Delta could be here and here,"},{"Start":"04:36.880 ","End":"04:38.725","Text":"or we\u0027d have to make it symmetrical."},{"Start":"04:38.725 ","End":"04:41.785","Text":"But it could be narrower."},{"Start":"04:41.785 ","End":"04:44.815","Text":"In fact, even if I took a smaller Delta,"},{"Start":"04:44.815 ","End":"04:46.255","Text":"that would still work."},{"Start":"04:46.255 ","End":"04:48.145","Text":"Delta is not unique."},{"Start":"04:48.145 ","End":"04:53.690","Text":"Anything smaller than a good Delta will also be a good Delta."},{"Start":"04:53.850 ","End":"04:56.230","Text":"Just note that in general,"},{"Start":"04:56.230 ","End":"04:58.360","Text":"if I make M larger,"},{"Start":"04:58.360 ","End":"05:00.965","Text":"suppose I took M over here,"},{"Start":"05:00.965 ","End":"05:05.230","Text":"then what would happen is that Delta would get smaller."},{"Start":"05:05.230 ","End":"05:08.840","Text":"We could take Delta perhaps here."},{"Start":"05:08.840 ","End":"05:10.760","Text":"As M gets larger and larger,"},{"Start":"05:10.760 ","End":"05:13.265","Text":"we have to take a smaller and smaller Delta."},{"Start":"05:13.265 ","End":"05:15.769","Text":"I\u0027ll give an example in a moment."},{"Start":"05:15.769 ","End":"05:20.810","Text":"What I\u0027d like to do first is give the definition for minus infinity,"},{"Start":"05:20.810 ","End":"05:23.450","Text":"it\u0027s actually infinity and minus infinity."},{"Start":"05:23.450 ","End":"05:26.145","Text":"Very similar but a bit different."},{"Start":"05:26.145 ","End":"05:27.860","Text":"Let\u0027s make the changes."},{"Start":"05:27.860 ","End":"05:32.395","Text":"Instead of infinity, we\u0027ll and have minus infinity."},{"Start":"05:32.395 ","End":"05:34.190","Text":"Then because it\u0027s minus infinity,"},{"Start":"05:34.190 ","End":"05:38.135","Text":"I\u0027ll change the word large to the word small."},{"Start":"05:38.135 ","End":"05:42.140","Text":"But the word small is a bit ambiguous."},{"Start":"05:42.140 ","End":"05:43.355","Text":"When I say small,"},{"Start":"05:43.355 ","End":"05:46.550","Text":"I mean very negative towards minus infinity,"},{"Start":"05:46.550 ","End":"05:49.775","Text":"not small as in close to 0."},{"Start":"05:49.775 ","End":"05:54.710","Text":"By making x close enough to a but not equal to a. It\u0027s the same."},{"Start":"05:54.710 ","End":"05:57.380","Text":"The difference here for the smallest we like,"},{"Start":"05:57.380 ","End":"06:01.165","Text":"instead of being bigger than some number M,"},{"Start":"06:01.165 ","End":"06:04.020","Text":"we want it to be below."},{"Start":"06:04.020 ","End":"06:05.340","Text":"I\u0027ll use a different latter,"},{"Start":"06:05.340 ","End":"06:13.285","Text":"I\u0027ll say smaller than N. I need to change this to an N,"},{"Start":"06:13.285 ","End":"06:16.550","Text":"I\u0027ll have to change with large,"},{"Start":"06:16.550 ","End":"06:21.290","Text":"however small, small meaning very, very negative."},{"Start":"06:21.290 ","End":"06:24.080","Text":"As for the picture, well,"},{"Start":"06:24.080 ","End":"06:26.525","Text":"I\u0027m not going to draw a fresh picture."},{"Start":"06:26.525 ","End":"06:32.975","Text":"Just imagine flipping it around the x-axis so that it\u0027s in the negative path."},{"Start":"06:32.975 ","End":"06:36.150","Text":"You know what? I\u0027ll just eliminate the picture."},{"Start":"06:37.540 ","End":"06:41.555","Text":"Now back to the infinity page."},{"Start":"06:41.555 ","End":"06:43.249","Text":"I\u0027m going to do an example."},{"Start":"06:43.249 ","End":"06:44.390","Text":"Let\u0027s see what I don\u0027t need."},{"Start":"06:44.390 ","End":"06:45.710","Text":"I don\u0027t need the picture,"},{"Start":"06:45.710 ","End":"06:47.380","Text":"I don\u0027t need the info."},{"Start":"06:47.380 ","End":"06:51.425","Text":"Now our example, what we\u0027ll do is take"},{"Start":"06:51.425 ","End":"07:00.195","Text":"the limit as x goes to 0 of 1 over x squared."},{"Start":"07:00.195 ","End":"07:04.685","Text":"We want to show that this is equal to infinity."},{"Start":"07:04.685 ","End":"07:06.600","Text":"I\u0027ll just highlight it,"},{"Start":"07:06.600 ","End":"07:08.790","Text":"so this is what we have to show."},{"Start":"07:08.790 ","End":"07:13.510","Text":"This is just like this except that we do know that a equals"},{"Start":"07:13.510 ","End":"07:20.570","Text":"0 and we know that f of x is 1 over x squared."},{"Start":"07:20.570 ","End":"07:23.340","Text":"Let\u0027s see what we have to show."},{"Start":"07:23.340 ","End":"07:25.990","Text":"This part\u0027s generic."},{"Start":"07:27.710 ","End":"07:34.650","Text":"There is a number Delta depending on M bigger than 0, such that,"},{"Start":"07:34.650 ","End":"07:41.060","Text":"this we have to change to 1 over x squared bigger"},{"Start":"07:41.060 ","End":"07:48.080","Text":"than M whenever 0 less than absolute value of x,"},{"Start":"07:48.080 ","End":"07:54.485","Text":"because a is 0, x minus 0 is just x less than Delta."},{"Start":"07:54.485 ","End":"07:57.445","Text":"I have to show that for each M,"},{"Start":"07:57.445 ","End":"08:03.215","Text":"there is a Delta such that this is true whenever this is true."},{"Start":"08:03.215 ","End":"08:06.590","Text":"Now what we usually do is we start with the,"},{"Start":"08:06.590 ","End":"08:09.110","Text":"now it used to be the Epsilon inequality."},{"Start":"08:09.110 ","End":"08:11.960","Text":"We start with the M inequality and see if we can"},{"Start":"08:11.960 ","End":"08:16.280","Text":"manipulate it to get something that looks a bit like this."},{"Start":"08:16.280 ","End":"08:21.160","Text":"What we can say, I\u0027ll start from here,"},{"Start":"08:21.160 ","End":"08:24.460","Text":"1 over x squared bigger than M,"},{"Start":"08:24.460 ","End":"08:28.280","Text":"I can change that."},{"Start":"08:28.460 ","End":"08:31.570","Text":"Now I can assume that M is positive."},{"Start":"08:31.570 ","End":"08:32.739","Text":"In fact, some definitions,"},{"Start":"08:32.739 ","End":"08:36.310","Text":"they say that M has to be positive."},{"Start":"08:37.130 ","End":"08:41.320","Text":"1 over x squared is bigger than M. Now why do I need it to be positive?"},{"Start":"08:41.320 ","End":"08:44.320","Text":"So I can manipulate the inequality and say that this"},{"Start":"08:44.320 ","End":"08:49.355","Text":"implies that 1 over M is bigger than x squared."},{"Start":"08:49.355 ","End":"08:56.264","Text":"Then if I take the square root of each side,"},{"Start":"08:56.264 ","End":"08:58.980","Text":"if M is positive, 1 over M is positive,"},{"Start":"08:58.980 ","End":"09:04.690","Text":"this gives me that 1 over"},{"Start":"09:04.690 ","End":"09:12.860","Text":"the square root of M is bigger than the square root of x squared is absolute value of x."},{"Start":"09:12.860 ","End":"09:15.050","Text":"If you\u0027re not sure about this direction,"},{"Start":"09:15.050 ","End":"09:19.834","Text":"certainly you can see that it\u0027s true in this direction that if this holds,"},{"Start":"09:19.834 ","End":"09:22.040","Text":"we have 2 positive numbers and we square them,"},{"Start":"09:22.040 ","End":"09:24.695","Text":"then this holds because really we\u0027re working backwards."},{"Start":"09:24.695 ","End":"09:28.250","Text":"At the end, we\u0027ll have to reach this,"},{"Start":"09:28.250 ","End":"09:30.500","Text":"we\u0027ll have to start with this and reach this."},{"Start":"09:30.500 ","End":"09:33.350","Text":"So it\u0027s a more important that this implies this."},{"Start":"09:33.350 ","End":"09:37.385","Text":"Now this gives us the idea because if we look at what we want to show,"},{"Start":"09:37.385 ","End":"09:44.160","Text":"we want to start off with x being less than Delta. Let\u0027s just try."},{"Start":"09:44.160 ","End":"09:48.720","Text":"We\u0027re going to try letting Delta equal 1 over"},{"Start":"09:48.720 ","End":"09:54.470","Text":"the square root of M. Now we have to start from beginning in the opposite direction."},{"Start":"09:54.470 ","End":"09:57.325","Text":"We have to prove that this whenever this,"},{"Start":"09:57.325 ","End":"10:01.530","Text":"which means that this has to imply this."},{"Start":"10:01.530 ","End":"10:06.120","Text":"This is the start and this is the end."},{"Start":"10:06.120 ","End":"10:07.950","Text":"If we start from,"},{"Start":"10:07.950 ","End":"10:10.175","Text":"this is what we have to show."},{"Start":"10:10.175 ","End":"10:16.610","Text":"We start from 0 less than absolute value of x less than Delta."},{"Start":"10:16.670 ","End":"10:19.845","Text":"Since Delta is 1 over M,"},{"Start":"10:19.845 ","End":"10:27.990","Text":"this implies that absolute value of x is less than 1 over"},{"Start":"10:27.990 ","End":"10:32.290","Text":"square root of M. I can just"},{"Start":"10:32.290 ","End":"10:36.775","Text":"ignore this part because if this is less than this less than this,"},{"Start":"10:36.775 ","End":"10:39.355","Text":"in particular, this is less than this."},{"Start":"10:39.355 ","End":"10:42.760","Text":"Now I can raise both sides to the power of 2."},{"Start":"10:42.760 ","End":"10:49.570","Text":"I can square both sides and get absolute value of x squared is"},{"Start":"10:49.570 ","End":"10:57.130","Text":"less than 1 over M. But absolute value of x squared is the same as x squared."},{"Start":"10:57.130 ","End":"11:00.440","Text":"So I have x squared is less than"},{"Start":"11:00.440 ","End":"11:05.950","Text":"1 over M. If I take the reciprocal of both sides I reverse the inequality."},{"Start":"11:05.950 ","End":"11:13.575","Text":"This gives me that 1 over x squared is bigger than 1 over 1 over M,"},{"Start":"11:13.575 ","End":"11:20.315","Text":"which is M. This is very good because we started with what we call start,"},{"Start":"11:20.315 ","End":"11:24.245","Text":"and we ended up with what we called N. We have shown this."},{"Start":"11:24.245 ","End":"11:30.135","Text":"That if we take Delta to be 1 over the square root of M,"},{"Start":"11:30.135 ","End":"11:33.015","Text":"then that will do the trick for us."},{"Start":"11:33.015 ","End":"11:35.900","Text":"I won\u0027t give an example with a minus infinity,"},{"Start":"11:35.900 ","End":"11:37.520","Text":"it\u0027s pretty much the same."},{"Start":"11:37.520 ","End":"11:39.860","Text":"But there is something else I\u0027d like to talk about"},{"Start":"11:39.860 ","End":"11:42.620","Text":"the 1-sided limits in the case of infinity."},{"Start":"11:42.620 ","End":"11:46.040","Text":"It works pretty much the same as in the cases as"},{"Start":"11:46.040 ","End":"11:49.710","Text":"if this was an L. Let\u0027s take the limit from the right."},{"Start":"11:49.710 ","End":"11:55.475","Text":"Suppose I wanted to have the same thing except x goes to a from above or from the right."},{"Start":"11:55.475 ","End":"12:00.935","Text":"All you\u0027d have to do would be to change this condition to 0"},{"Start":"12:00.935 ","End":"12:08.285","Text":"less than x minus a less than Delta instead of the absolute value."},{"Start":"12:08.285 ","End":"12:13.580","Text":"This of course, can be written also as x between"},{"Start":"12:13.580 ","End":"12:20.880","Text":"a and a plus Delta."},{"Start":"12:21.050 ","End":"12:26.795","Text":"For the case of the limit from the left or from below,"},{"Start":"12:26.795 ","End":"12:31.115","Text":"x goes to a from the left."},{"Start":"12:31.115 ","End":"12:39.770","Text":"We just replace this by minus Delta less than x minus a less than 0,"},{"Start":"12:39.770 ","End":"12:42.365","Text":"or if you prefer it in the other form,"},{"Start":"12:42.365 ","End":"12:48.225","Text":"it\u0027s that a minus Delta less than x,"},{"Start":"12:48.225 ","End":"12:54.630","Text":"less than a. Yeah,"},{"Start":"12:54.630 ","End":"12:58.670","Text":"this means to the right of a with up to a distance of Delta,"},{"Start":"12:58.670 ","End":"13:02.965","Text":"and this means to the left of a up to a distance of Delta."},{"Start":"13:02.965 ","End":"13:05.660","Text":"Similarly, for minus infinity,"},{"Start":"13:05.660 ","End":"13:07.130","Text":"I\u0027m not going to do the same."},{"Start":"13:07.130 ","End":"13:11.120","Text":"Exactly the same change would work in the case of the minus infinity."},{"Start":"13:11.120 ","End":"13:15.340","Text":"They\u0027re also both very similar to the case where it\u0027s just an L here,"},{"Start":"13:15.340 ","End":"13:23.860","Text":"just a matter of changing this condition to separate conditions without absolute value."},{"Start":"13:24.020 ","End":"13:29.140","Text":"That is all for the infinite limits meanwhile."}],"ID":8307},{"Watched":false,"Name":"Limit at Infinity","Duration":"10m 24s","ChapterTopicVideoID":8154,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"Now we come to the definition of the limit of a function at infinity."},{"Start":"00:04.575 ","End":"00:07.260","Text":"Unlike the previous clip where we had"},{"Start":"00:07.260 ","End":"00:10.980","Text":"an infinite limit and the different concepts as you\u0027ll soon see."},{"Start":"00:10.980 ","End":"00:16.030","Text":"Let\u0027s start out with the regular definition of a limit in the most basic case,"},{"Start":"00:16.030 ","End":"00:19.475","Text":"where x goes to a and the limit is some"},{"Start":"00:19.475 ","End":"00:24.170","Text":"L. The infinite limit is when we replaced L by infinity."},{"Start":"00:24.170 ","End":"00:28.130","Text":"Now we\u0027re going to do something different and replace a by infinity."},{"Start":"00:28.130 ","End":"00:31.285","Text":"Instead of this a, I\u0027m going to put infinity,"},{"Start":"00:31.285 ","End":"00:33.860","Text":"and later on we\u0027ll also deal with minus infinity,"},{"Start":"00:33.860 ","End":"00:36.200","Text":"but let\u0027s just stick to infinity for the moment."},{"Start":"00:36.200 ","End":"00:38.749","Text":"Let\u0027s see if we can translate the definition,"},{"Start":"00:38.749 ","End":"00:41.000","Text":"see what it would be in the case of infinity."},{"Start":"00:41.000 ","End":"00:44.020","Text":"Let\u0027s start off with the informal definition."},{"Start":"00:44.020 ","End":"00:47.870","Text":"We can make f of x as close as we like to L. Well,"},{"Start":"00:47.870 ","End":"00:51.155","Text":"there\u0027s no problem there because L is a regular number."},{"Start":"00:51.155 ","End":"00:55.170","Text":"By making x close enough to a."},{"Start":"00:55.170 ","End":"00:56.590","Text":"Now a is infinity,"},{"Start":"00:56.590 ","End":"00:59.515","Text":"now what do we mean by close enough to infinity?"},{"Start":"00:59.515 ","End":"01:02.110","Text":"The not equal to a,"},{"Start":"01:02.110 ","End":"01:06.330","Text":"we can immediately erase because x is not equal to infinity."},{"Start":"01:06.330 ","End":"01:10.780","Text":"But what do we mean by close enough to infinity?"},{"Start":"01:10.780 ","End":"01:17.550","Text":"Well, close enough to infinity means very, very large basically."},{"Start":"01:17.550 ","End":"01:21.235","Text":"See if we can translate that into a definition here."},{"Start":"01:21.235 ","End":"01:30.195","Text":"Now, the Epsilon part will be the same but what do we do with this here?"},{"Start":"01:30.195 ","End":"01:32.450","Text":"This deals with a finite quantity a,"},{"Start":"01:32.450 ","End":"01:34.695","Text":"when we\u0027re close to a finite a."},{"Start":"01:34.695 ","End":"01:36.820","Text":"When a is infinity, like I said,"},{"Start":"01:36.820 ","End":"01:40.805","Text":"it just means that it\u0027s very, very big."},{"Start":"01:40.805 ","End":"01:43.500","Text":"I then want Delta."},{"Start":"01:43.500 ","End":"01:45.380","Text":"Delta usually means a small number."},{"Start":"01:45.380 ","End":"01:47.665","Text":"I want something very large."},{"Start":"01:47.665 ","End":"01:50.160","Text":"I\u0027ll call that M,"},{"Start":"01:50.160 ","End":"01:53.585","Text":"such that f of x is close to L whenever,"},{"Start":"01:53.585 ","End":"01:58.550","Text":"and this has to be replaced by x close to"},{"Start":"01:58.550 ","End":"02:05.120","Text":"infinity means x bigger than M and we\u0027ll see an example of that."},{"Start":"02:05.120 ","End":"02:08.290","Text":"There\u0027s one other thing though about the domain."},{"Start":"02:08.290 ","End":"02:14.510","Text":"We can\u0027t leave this as an interval containing a when a is infinity."},{"Start":"02:14.510 ","End":"02:19.985","Text":"For one thing, we don\u0027t have something at infinity itself. It\u0027s not a point."},{"Start":"02:19.985 ","End":"02:23.165","Text":"But instead of an interval containing a,"},{"Start":"02:23.165 ","End":"02:27.110","Text":"we have an interval containing infinity,"},{"Start":"02:27.110 ","End":"02:31.090","Text":"which means x larger than something."},{"Start":"02:31.090 ","End":"02:33.575","Text":"That\u0027s where we considered the analog of"},{"Start":"02:33.575 ","End":"02:37.480","Text":"an interval containing infinity from some point onwards."},{"Start":"02:37.480 ","End":"02:40.265","Text":"We call that interval x bigger than,"},{"Start":"02:40.265 ","End":"02:47.330","Text":"let\u0027s say for some K. If f is defined from some point onwards,"},{"Start":"02:47.330 ","End":"02:49.925","Text":"you could think of this as the interval K,"},{"Start":"02:49.925 ","End":"02:52.040","Text":"infinity if you like it better."},{"Start":"02:52.040 ","End":"02:55.385","Text":"Here\u0027s a little sketch that might help explain things."},{"Start":"02:55.385 ","End":"02:57.335","Text":"K doesn\u0027t appear in the sketch,"},{"Start":"02:57.335 ","End":"03:03.270","Text":"but the function is defined from some point on all the way up to infinity."},{"Start":"03:03.640 ","End":"03:08.555","Text":"Here we see the limit L,"},{"Start":"03:08.555 ","End":"03:13.860","Text":"which is hinted at that the function which is in red,"},{"Start":"03:13.860 ","End":"03:16.480","Text":"when it goes larger and larger towards infinity,"},{"Start":"03:16.480 ","End":"03:17.740","Text":"it seems to stabilize."},{"Start":"03:17.740 ","End":"03:22.300","Text":"It seems to approach this value L. What we require is that if we\u0027re"},{"Start":"03:22.300 ","End":"03:27.730","Text":"given Epsilon and we mark an interval plus or minus Epsilon,"},{"Start":"03:27.730 ","End":"03:30.080","Text":"from some point onwards,"},{"Start":"03:30.080 ","End":"03:31.580","Text":"in this case, M,"},{"Start":"03:31.580 ","End":"03:35.620","Text":"the function is entirely within this band of"},{"Start":"03:35.620 ","End":"03:39.870","Text":"width Epsilon around L. We can take Epsilon as small as we want,"},{"Start":"03:39.870 ","End":"03:44.975","Text":"so the function is forced to get closer and closer to this line,"},{"Start":"03:44.975 ","End":"03:51.305","Text":"y equals L as we get further and further towards infinity or to the right."},{"Start":"03:51.305 ","End":"03:53.855","Text":"Anyway, that\u0027s just an illustration."},{"Start":"03:53.855 ","End":"03:58.110","Text":"Now let\u0027s look at the case of minus infinity."},{"Start":"03:59.210 ","End":"04:02.930","Text":"Let\u0027s see what we have to change."},{"Start":"04:02.930 ","End":"04:08.614","Text":"It\u0027s defined for x less than K all the way down to minus infinity."},{"Start":"04:08.614 ","End":"04:16.995","Text":"I can write this as minus infinity to K if you prefer the interval notation."},{"Start":"04:16.995 ","End":"04:23.300","Text":"This definition with the minus and what we have is the Epsilon part"},{"Start":"04:23.300 ","End":"04:29.270","Text":"stays the same but yeah,"},{"Start":"04:29.270 ","End":"04:30.740","Text":"let\u0027s go down to the informal."},{"Start":"04:30.740 ","End":"04:37.700","Text":"Close enough to minus infinity means very,"},{"Start":"04:37.700 ","End":"04:41.345","Text":"very small in the sense of very, very negative."},{"Start":"04:41.345 ","End":"04:45.244","Text":"For each Epsilon, there is an M less than 0,"},{"Start":"04:45.244 ","End":"04:48.965","Text":"such that whenever x is very close to L,"},{"Start":"04:48.965 ","End":"04:55.640","Text":"then x is less than M. That\u0027s how the definition changes."},{"Start":"04:55.640 ","End":"04:58.730","Text":"The picture, I don\u0027t have one for minus infinity."},{"Start":"04:58.730 ","End":"05:01.595","Text":"I just deleted it so it won\u0027t be confusing."},{"Start":"05:01.595 ","End":"05:04.040","Text":"Now an example."},{"Start":"05:04.040 ","End":"05:10.070","Text":"Let\u0027s take as an example to show that"},{"Start":"05:10.070 ","End":"05:19.280","Text":"the limit as x goes to infinity of 1/x is equal to,"},{"Start":"05:19.280 ","End":"05:20.945","Text":"can you guess it?"},{"Start":"05:20.945 ","End":"05:26.810","Text":"0. It\u0027s like we\u0027re given"},{"Start":"05:26.810 ","End":"05:32.660","Text":"Epsilon and we have to find M with the property that this implies this."},{"Start":"05:32.660 ","End":"05:34.190","Text":"When I say this, whenever this,"},{"Start":"05:34.190 ","End":"05:36.485","Text":"it means that the arrow goes in this direction."},{"Start":"05:36.485 ","End":"05:45.510","Text":"It means that whenever x is bigger than M then absolute value of f of x."},{"Start":"05:45.510 ","End":"05:54.435","Text":"Well, f of x is 1/x and L is 0,"},{"Start":"05:54.435 ","End":"05:58.965","Text":"is going to be less than Epsilon."},{"Start":"05:58.965 ","End":"06:02.310","Text":"Now, this is absolute value looks like a 1."},{"Start":"06:02.310 ","End":"06:08.680","Text":"Yeah. Now M is a big number,"},{"Start":"06:08.680 ","End":"06:11.965","Text":"and like I said before, it\u0027s okay to assume M is bigger than 0,"},{"Start":"06:11.965 ","End":"06:16.015","Text":"in which case, 1/x is also bigger than 0."},{"Start":"06:16.015 ","End":"06:22.940","Text":"This just comes out to be 1/x less than Epsilon."},{"Start":"06:22.940 ","End":"06:27.370","Text":"This means that if I just switch x and Epsilon,"},{"Start":"06:27.370 ","End":"06:30.365","Text":"or if I take the reciprocal of both sides, maybe that\u0027s easier,"},{"Start":"06:30.365 ","End":"06:33.965","Text":"then x is bigger than 1/Epsilon."},{"Start":"06:33.965 ","End":"06:40.145","Text":"Now we\u0027re looking for an x bigger than M. We\u0027re going to try or candidate for"},{"Start":"06:40.145 ","End":"06:47.175","Text":"M is that M of Epsilon is equal to 1/Epsilon."},{"Start":"06:47.175 ","End":"06:50.570","Text":"Now we have to do it the right way round to start from"},{"Start":"06:50.570 ","End":"06:55.345","Text":"here and to get here to make sure that it\u0027s okay."},{"Start":"06:55.345 ","End":"06:57.770","Text":"Well, I guess we\u0027ll go over here, yeah."},{"Start":"06:57.770 ","End":"07:00.200","Text":"Let\u0027s assume that x is bigger than M"},{"Start":"07:00.200 ","End":"07:02.900","Text":"and I have to do a series of deductions and get to this."},{"Start":"07:02.900 ","End":"07:05.615","Text":"If x is bigger than M,"},{"Start":"07:05.615 ","End":"07:09.680","Text":"but M is equal to 1/Epsilon,"},{"Start":"07:09.680 ","End":"07:13.000","Text":"then x is bigger than 1/Epsilon."},{"Start":"07:13.000 ","End":"07:19.010","Text":"Taking the reciprocals, we have 1/x is less than the reciprocal of this,"},{"Start":"07:19.010 ","End":"07:22.780","Text":"which is 1/Epsilon, which is Epsilon."},{"Start":"07:22.780 ","End":"07:27.650","Text":"This means that since x is positive,"},{"Start":"07:27.650 ","End":"07:30.825","Text":"that 1/x minus 0,"},{"Start":"07:30.825 ","End":"07:33.345","Text":"same thing is less than Epsilon."},{"Start":"07:33.345 ","End":"07:36.030","Text":"This is just what is written here."},{"Start":"07:36.030 ","End":"07:43.070","Text":"Looks like we\u0027ve done it and if we take our M to be 1/Epsilon here,"},{"Start":"07:43.070 ","End":"07:48.270","Text":"then it satisfies the condition that whenever this is true, this is true."},{"Start":"07:48.710 ","End":"07:52.100","Text":"That\u0027s it. I won\u0027t do an example for minus infinity."},{"Start":"07:52.100 ","End":"07:54.050","Text":"It\u0027s almost the same thing."},{"Start":"07:54.050 ","End":"07:56.090","Text":"What I do want though,"},{"Start":"07:56.090 ","End":"07:59.060","Text":"there\u0027s one more thing in terms of infinity."},{"Start":"07:59.060 ","End":"08:03.680","Text":"What I want to do now is introduce what I call a hybrid case."},{"Start":"08:03.680 ","End":"08:06.560","Text":"There were two kinds of infinity limits."},{"Start":"08:06.560 ","End":"08:10.020","Text":"One is the limit at infinity is like here,"},{"Start":"08:10.020 ","End":"08:12.245","Text":"when we let x go to infinity,"},{"Start":"08:12.245 ","End":"08:15.890","Text":"and previously we had that the limit was infinity,"},{"Start":"08:15.890 ","End":"08:18.545","Text":"that x went to some number a,"},{"Start":"08:18.545 ","End":"08:20.510","Text":"but the limit was infinity."},{"Start":"08:20.510 ","End":"08:25.940","Text":"Now I want to combine them both into the case"},{"Start":"08:25.940 ","End":"08:31.565","Text":"where x goes to infinity and the limit is infinity."},{"Start":"08:31.565 ","End":"08:36.990","Text":"I\u0027m going to modify this to make this infinity."},{"Start":"08:36.990 ","End":"08:40.970","Text":"The way it works is we have to modify this."},{"Start":"08:40.970 ","End":"08:42.575","Text":"The Epsilon doesn\u0027t belong,"},{"Start":"08:42.575 ","End":"08:44.465","Text":"the M part is okay."},{"Start":"08:44.465 ","End":"08:47.910","Text":"The definition is that,"},{"Start":"08:49.840 ","End":"08:53.690","Text":"I\u0027ll change this Epsilon to an N,"},{"Start":"08:53.690 ","End":"08:55.700","Text":"so for every number N bigger than 0,"},{"Start":"08:55.700 ","End":"08:59.600","Text":"there is another number M bigger than 0 it may depend on N,"},{"Start":"08:59.600 ","End":"09:04.770","Text":"such that f of x is bigger than N"},{"Start":"09:04.770 ","End":"09:10.770","Text":"whenever x is bigger than M. This is the definition for this."},{"Start":"09:10.770 ","End":"09:13.310","Text":"There\u0027s actually four of these."},{"Start":"09:13.310 ","End":"09:15.320","Text":"Is this and three more because this could be"},{"Start":"09:15.320 ","End":"09:20.030","Text":"plus or minus infinity and this"},{"Start":"09:20.030 ","End":"09:25.490","Text":"could be plus or minus infinity and they all work the same way."},{"Start":"09:25.490 ","End":"09:30.540","Text":"Just the signs on the inequalities here."},{"Start":"09:32.960 ","End":"09:38.720","Text":"For example, if I make this less than 0 and this less than 0,"},{"Start":"09:38.720 ","End":"09:44.510","Text":"then it\u0027s the minus infinity here and the plus infinity here."},{"Start":"09:44.510 ","End":"09:48.815","Text":"If I did it the other way around and I switched this and this,"},{"Start":"09:48.815 ","End":"09:51.450","Text":"then it would be plus infinity and minus infinity."},{"Start":"09:51.450 ","End":"09:52.760","Text":"If I switched them both,"},{"Start":"09:52.760 ","End":"09:54.305","Text":"then it\u0027s minus and minus."},{"Start":"09:54.305 ","End":"09:57.035","Text":"I\u0027m not going to do all the other cases."},{"Start":"09:57.035 ","End":"09:58.850","Text":"They\u0027re just, as I said,"},{"Start":"09:58.850 ","End":"10:03.320","Text":"just a matter of switching the signs of"},{"Start":"10:03.320 ","End":"10:06.650","Text":"the inequalities to the four possible combinations"},{"Start":"10:06.650 ","End":"10:10.510","Text":"and we get the three other definitions."},{"Start":"10:10.510 ","End":"10:15.300","Text":"I believe there\u0027s at least one of these cases in"},{"Start":"10:15.300 ","End":"10:21.185","Text":"the solved examples following the tutorial so take a look there."},{"Start":"10:21.185 ","End":"10:24.960","Text":"Other than that, I\u0027m done with this clip."}],"ID":8308},{"Watched":false,"Name":"Definition of Continuity","Duration":"4m 59s","ChapterTopicVideoID":8155,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.270","Text":"I\u0027m going to talk about the definition of continuity of a function at a point."},{"Start":"00:06.270 ","End":"00:09.165","Text":"This time the Epsilon Delta definition."},{"Start":"00:09.165 ","End":"00:13.425","Text":"Up \u0027til now we\u0027ve been vaguely defining continuity,"},{"Start":"00:13.425 ","End":"00:19.710","Text":"and we\u0027re going to use the definition of a limit using Epsilon Delta."},{"Start":"00:19.710 ","End":"00:22.355","Text":"If instead of continuity,"},{"Start":"00:22.355 ","End":"00:24.950","Text":"I wrote the limit,"},{"Start":"00:24.950 ","End":"00:27.270","Text":"then this is what we did already."},{"Start":"00:27.270 ","End":"00:31.160","Text":"Let\u0027s take a look at 1 of the previous clips."},{"Start":"00:31.160 ","End":"00:36.455","Text":"Now, the definition of a limit is very close to the definition of continuity."},{"Start":"00:36.455 ","End":"00:41.495","Text":"In the case of a limit, we have some letter L on the right, some number,"},{"Start":"00:41.495 ","End":"00:43.190","Text":"and in the case of continuity,"},{"Start":"00:43.190 ","End":"00:47.195","Text":"if we just replaced this L by f of a, that\u0027s what we\u0027ll get."},{"Start":"00:47.195 ","End":"00:52.275","Text":"Let me just borrow this and copy paste it."},{"Start":"00:52.275 ","End":"00:55.395","Text":"Back here and now paste."},{"Start":"00:55.395 ","End":"00:59.655","Text":"Now let\u0027s see how we modify it, for one thing,"},{"Start":"00:59.655 ","End":"01:02.240","Text":"if you\u0027re talking about continuity,"},{"Start":"01:02.240 ","End":"01:04.640","Text":"at a point, say x equals a,"},{"Start":"01:04.640 ","End":"01:06.890","Text":"we need the function to be defined there."},{"Start":"01:06.890 ","End":"01:10.130","Text":"This extra bit can go."},{"Start":"01:10.130 ","End":"01:15.995","Text":"We\u0027re talking about a function that\u0027s defined on an interval containing our point a."},{"Start":"01:15.995 ","End":"01:19.190","Text":"Now, we had an old definition of continuity,"},{"Start":"01:19.190 ","End":"01:25.220","Text":"instead of the L here, I just erased it, we put f of a."},{"Start":"01:25.220 ","End":"01:28.339","Text":"I\u0027ll just change the wording."},{"Start":"01:28.339 ","End":"01:31.310","Text":"This is the old definition we had then we said"},{"Start":"01:31.310 ","End":"01:40.110","Text":"that f is continuous at x equals a,"},{"Start":"01:40.110 ","End":"01:42.540","Text":"I forgot the word \"is\","},{"Start":"01:42.540 ","End":"01:46.830","Text":"continuous at x equals a if, like I said,"},{"Start":"01:46.830 ","End":"01:48.270","Text":"this was the old definition,"},{"Start":"01:48.270 ","End":"01:49.550","Text":"limit as x goes to a,"},{"Start":"01:49.550 ","End":"01:51.920","Text":"f of x is f of a."},{"Start":"01:51.920 ","End":"01:55.810","Text":"Now we\u0027ll have a new definition with Epsilon Delta,"},{"Start":"01:55.810 ","End":"01:58.595","Text":"so the Epsilon Delta definition,"},{"Start":"01:58.595 ","End":"02:02.210","Text":"It\u0027s the same as what we had before when we had L here,"},{"Start":"02:02.210 ","End":"02:04.160","Text":"but we replace L by f of x,"},{"Start":"02:04.160 ","End":"02:10.690","Text":"so this L goes and we write f of a."},{"Start":"02:10.730 ","End":"02:14.900","Text":"For every number Epsilon bigger than 0,"},{"Start":"02:14.900 ","End":"02:16.145","Text":"no matter how small,"},{"Start":"02:16.145 ","End":"02:18.850","Text":"we can find another number Delta,"},{"Start":"02:18.850 ","End":"02:22.380","Text":"such that f of x will be close to f of a within"},{"Start":"02:22.380 ","End":"02:27.115","Text":"Epsilon whenever x is close to a within Delta."},{"Start":"02:27.115 ","End":"02:32.345","Text":"In this case we can actually eliminate the bigger than 0."},{"Start":"02:32.345 ","End":"02:39.035","Text":"This is no longer important because if x minus a is 0,"},{"Start":"02:39.035 ","End":"02:41.060","Text":"then x is equal to a,"},{"Start":"02:41.060 ","End":"02:44.810","Text":"and certainly f of x will be then f of a and this will be 0,"},{"Start":"02:44.810 ","End":"02:46.535","Text":"which will be less than Epsilon."},{"Start":"02:46.535 ","End":"02:56.535","Text":"This will be our definition of f is continuous at x equals a."},{"Start":"02:56.535 ","End":"03:02.735","Text":"Where as I said, f is defined in some interval containing x equals a and that a itself,"},{"Start":"03:02.735 ","End":"03:05.130","Text":"which is unlike before."},{"Start":"03:06.800 ","End":"03:10.810","Text":"Why don\u0027t I put this in a box?"},{"Start":"03:11.210 ","End":"03:13.940","Text":"Now, I don\u0027t really need to give"},{"Start":"03:13.940 ","End":"03:16.730","Text":"examples because it turns out that the only examples we\u0027ve"},{"Start":"03:16.730 ","End":"03:21.575","Text":"done earlier of limits can also be used for continuity."},{"Start":"03:21.575 ","End":"03:24.635","Text":"But let me just take one of them just to show you what I mean."},{"Start":"03:24.635 ","End":"03:30.304","Text":"Suppose I asked you to show that f of x equals x squared"},{"Start":"03:30.304 ","End":"03:38.235","Text":"is continuous at x equals 0."},{"Start":"03:38.235 ","End":"03:43.520","Text":"What we would have to show basically is this,"},{"Start":"03:43.520 ","End":"03:45.260","Text":"but in Epsilon Delta."},{"Start":"03:45.260 ","End":"03:52.160","Text":"This would mean that the limit as x goes to 0 of x"},{"Start":"03:52.160 ","End":"04:00.245","Text":"squared equals f of 0 is 0 squared is 0."},{"Start":"04:00.245 ","End":"04:02.810","Text":"But we\u0027ve already done this."},{"Start":"04:02.810 ","End":"04:05.810","Text":"There\u0027s a quick flash back to the lesson where we"},{"Start":"04:05.810 ","End":"04:08.630","Text":"showed that we proved already that the limit of"},{"Start":"04:08.630 ","End":"04:14.615","Text":"x squared as x goes to 0 is 0 using Epsilon Delta if you follow this."},{"Start":"04:14.615 ","End":"04:17.030","Text":"Since we\u0027ve done this,"},{"Start":"04:17.030 ","End":"04:24.710","Text":"and we have this and therefore it\u0027s continuous because 0 is f of 0,"},{"Start":"04:24.710 ","End":"04:29.685","Text":"this is f of x and this is f of 0."},{"Start":"04:29.685 ","End":"04:31.775","Text":"If you look at all the other examples,"},{"Start":"04:31.775 ","End":"04:35.930","Text":"all the other limits are also proofs of continuity."},{"Start":"04:35.930 ","End":"04:40.160","Text":"Because in all cases the L here was in fact"},{"Start":"04:40.160 ","End":"04:46.020","Text":"the value of the function at the point that we were taking the limit to."},{"Start":"04:46.800 ","End":"04:52.970","Text":"That\u0027s it for theory and there are solved"},{"Start":"04:52.970 ","End":"04:57.560","Text":"examples after the tutorial and you\u0027ll learn more from those."},{"Start":"04:57.560 ","End":"04:59.730","Text":"We\u0027re done."}],"ID":8309},{"Watched":false,"Name":"Exercise 1","Duration":"2m 50s","ChapterTopicVideoID":8157,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we\u0027re going to use the definition of the limit,"},{"Start":"00:03.450 ","End":"00:07.050","Text":"and I\u0027m talking about the Epsilon-Delta definition"},{"Start":"00:07.050 ","End":"00:11.100","Text":"to prove that this limit holds."},{"Start":"00:11.100 ","End":"00:15.180","Text":"I just copied this and identified the 2 is a,"},{"Start":"00:15.180 ","End":"00:17.745","Text":"the 7x plus 14 as f of x,"},{"Start":"00:17.745 ","End":"00:19.170","Text":"and the 28 is L."},{"Start":"00:19.170 ","End":"00:21.960","Text":"We have a definition,"},{"Start":"00:21.960 ","End":"00:24.255","Text":"and here it is again."},{"Start":"00:24.255 ","End":"00:28.055","Text":"In general, we\u0027re given an Epsilon and we have to provide a Delta."},{"Start":"00:28.055 ","End":"00:32.120","Text":"The Delta we have to provide is such that this implies this."},{"Start":"00:32.120 ","End":"00:38.195","Text":"Whenever x is close to a within Delta but not equal to a,"},{"Start":"00:38.195 ","End":"00:41.240","Text":"then f of x is close to L within Epsilon."},{"Start":"00:41.240 ","End":"00:44.090","Text":"That\u0027s what this basically says with absolute values."},{"Start":"00:44.090 ","End":"00:46.485","Text":"We usually start with this,"},{"Start":"00:46.485 ","End":"00:48.395","Text":"with the Epsilon condition,"},{"Start":"00:48.395 ","End":"00:57.010","Text":"and we can write it as absolute value of f of x is 7x plus 14 minus L,"},{"Start":"00:57.010 ","End":"01:00.855","Text":"which is 28 less than Epsilon."},{"Start":"01:00.855 ","End":"01:05.720","Text":"Let\u0027s see if we can get to a condition on x minus a,"},{"Start":"01:05.720 ","End":"01:07.910","Text":"which is x minus 2."},{"Start":"01:07.910 ","End":"01:10.100","Text":"Simplifying what\u0027s inside,"},{"Start":"01:10.100 ","End":"01:14.070","Text":"this is 7x plus 14 minus 28,"},{"Start":"01:14.070 ","End":"01:20.065","Text":"so it\u0027s 7x minus 14 less than Epsilon."},{"Start":"01:20.065 ","End":"01:23.355","Text":"We can take 7 outside the brackets."},{"Start":"01:23.355 ","End":"01:24.680","Text":"Because 7 is positive,"},{"Start":"01:24.680 ","End":"01:27.100","Text":"we can take it outside the absolute value,"},{"Start":"01:27.100 ","End":"01:32.335","Text":"so we get 7 times x minus 2 is less than Epsilon."},{"Start":"01:32.335 ","End":"01:34.510","Text":"If we divide by 7,"},{"Start":"01:34.510 ","End":"01:40.160","Text":"we\u0027ve got x minus 2 in absolute value is less than Epsilon over 7."},{"Start":"01:40.160 ","End":"01:44.465","Text":"Now, this looks quite a bit like this."},{"Start":"01:44.465 ","End":"01:46.550","Text":"What we\u0027re going to do is try letting"},{"Start":"01:46.550 ","End":"01:50.420","Text":"Delta equals Epsilon over 7"},{"Start":"01:50.420 ","End":"01:55.890","Text":"and see if we can get the other way from here to here."},{"Start":"01:55.990 ","End":"02:04.550","Text":"Well, certainly, if x minus a is bigger than 0 and less than Delta,"},{"Start":"02:04.550 ","End":"02:08.820","Text":"we can ignore the bigger than 0 part, it\u0027s less than Delta."},{"Start":"02:09.200 ","End":"02:11.840","Text":"This is equal to Delta now,"},{"Start":"02:11.840 ","End":"02:15.455","Text":"so if x minus 2 is less than Delta,"},{"Start":"02:15.455 ","End":"02:19.160","Text":"then we can do the steps in reverse."},{"Start":"02:19.160 ","End":"02:22.055","Text":"Then this is true because multiply by 7,"},{"Start":"02:22.055 ","End":"02:25.400","Text":"put the 7 inside and rewrite this this way."},{"Start":"02:25.400 ","End":"02:27.800","Text":"In other words, if this is true,"},{"Start":"02:27.800 ","End":"02:30.905","Text":"then this is true because all the steps work backwards."},{"Start":"02:30.905 ","End":"02:33.170","Text":"I could write it again also just to be clear that"},{"Start":"02:33.170 ","End":"02:38.460","Text":"if x minus 2 is less than Delta,"},{"Start":"02:38.460 ","End":"02:41.480","Text":"where Delta is equal to Epsilon over 7,"},{"Start":"02:41.480 ","End":"02:45.875","Text":"we could get after going backwards to this,"},{"Start":"02:45.875 ","End":"02:47.885","Text":"which is what we wanted."},{"Start":"02:47.885 ","End":"02:50.850","Text":"That\u0027s it. We\u0027re done."}],"ID":8311},{"Watched":false,"Name":"Exercise 2","Duration":"5m 3s","ChapterTopicVideoID":8158,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this exercise, we have to prove that this limit holds"},{"Start":"00:03.150 ","End":"00:04.350","Text":"using the definition,"},{"Start":"00:04.350 ","End":"00:08.580","Text":"and I\u0027m referring to the Epsilon-Delta definition."},{"Start":"00:08.580 ","End":"00:11.220","Text":"I\u0027ll remind you of the rule."},{"Start":"00:11.220 ","End":"00:14.354","Text":"This is the familiar rule in general."},{"Start":"00:14.354 ","End":"00:17.250","Text":"We have to show that to each Epsilon,"},{"Start":"00:17.250 ","End":"00:18.300","Text":"there\u0027s a Delta,"},{"Start":"00:18.300 ","End":"00:20.910","Text":"such that whenever this is true, then this is true,"},{"Start":"00:20.910 ","End":"00:23.280","Text":"the implication goes from here to here,"},{"Start":"00:23.280 ","End":"00:26.145","Text":"and in our case, we have a is 3,"},{"Start":"00:26.145 ","End":"00:29.385","Text":"f of x is x squared and L is 9."},{"Start":"00:29.385 ","End":"00:35.415","Text":"Now we usually start from this condition and work our way back to this condition,"},{"Start":"00:35.415 ","End":"00:41.220","Text":"so f of x is x squared and L is 9."},{"Start":"00:41.220 ","End":"00:45.445","Text":"We know that this is less than Epsilon"},{"Start":"00:45.445 ","End":"00:51.870","Text":"and we have to work our way backwards to some condition like this."},{"Start":"00:52.000 ","End":"00:57.170","Text":"Now we can factorize this using difference of squares formulas"},{"Start":"00:57.170 ","End":"01:02.750","Text":"as the absolute value of x minus 3"},{"Start":"01:02.750 ","End":"01:06.050","Text":"times absolute value of x plus 3"},{"Start":"01:06.050 ","End":"01:08.390","Text":"using the a squared minus b squared formula,"},{"Start":"01:08.390 ","End":"01:10.280","Text":"a minus b, a plus b."},{"Start":"01:10.280 ","End":"01:16.145","Text":"Now this is good because we have an x minus 3 here on which is what is here,"},{"Start":"01:16.145 ","End":"01:18.725","Text":"because our a is 3."},{"Start":"01:18.725 ","End":"01:22.730","Text":"In an earlier exercise, we had something like this."},{"Start":"01:22.730 ","End":"01:26.630","Text":"Only here, we had a constant like 7,"},{"Start":"01:26.630 ","End":"01:30.460","Text":"and if this was a constant like 7"},{"Start":"01:30.460 ","End":"01:32.400","Text":"and this was less than Epsilon,"},{"Start":"01:32.400 ","End":"01:37.235","Text":"then we could divide by the 7 and say that this was less than Epsilon over 7."},{"Start":"01:37.235 ","End":"01:41.145","Text":"What we can do is make an estimate on this."},{"Start":"01:41.145 ","End":"01:43.010","Text":"Here\u0027s how we proceed."},{"Start":"01:43.010 ","End":"01:52.310","Text":"We make a first estimate on x minus 3 and we can say that it\u0027s less than 1."},{"Start":"01:52.310 ","End":"01:54.680","Text":"1 is a typical number to take."},{"Start":"01:54.680 ","End":"01:57.500","Text":"I can always make my Delta smaller than 1"},{"Start":"01:57.500 ","End":"02:00.260","Text":"because when you make Delta smaller, it also works."},{"Start":"02:00.260 ","End":"02:04.130","Text":"I\u0027ll even write that Delta is less than 1,"},{"Start":"02:04.130 ","End":"02:06.650","Text":"then x minus 3 less than 1."},{"Start":"02:06.650 ","End":"02:09.170","Text":"But what about x plus 3 now?"},{"Start":"02:09.170 ","End":"02:13.144","Text":"We can do some algebra with absolute values."},{"Start":"02:13.144 ","End":"02:21.505","Text":"This inequality means that the distance of x from 3 is less than 1,"},{"Start":"02:21.505 ","End":"02:24.670","Text":"which means that x is between,"},{"Start":"02:24.670 ","End":"02:30.605","Text":"3 plus 1 is 4 and 3 minus 1 is 2."},{"Start":"02:30.605 ","End":"02:33.739","Text":"That means that if I add 3,"},{"Start":"02:33.739 ","End":"02:40.345","Text":"then x plus 3 is between 7 and 5."},{"Start":"02:40.345 ","End":"02:42.770","Text":"Of course, if a number is between 5 and 7,"},{"Start":"02:42.770 ","End":"02:46.340","Text":"it\u0027s positive and I can replace it with the absolute value,"},{"Start":"02:46.340 ","End":"02:48.145","Text":"so that won\u0027t hurt."},{"Start":"02:48.145 ","End":"02:52.070","Text":"Provided that we know that x minus 3 is less than 1,"},{"Start":"02:52.070 ","End":"02:55.340","Text":"then we have that this is between 5 and 7."},{"Start":"02:55.340 ","End":"02:56.435","Text":"The question is,"},{"Start":"02:56.435 ","End":"02:59.240","Text":"which would I take to replace the x plus 3?"},{"Start":"02:59.240 ","End":"03:01.970","Text":"Do I take the 5 or do I take the 7?"},{"Start":"03:01.970 ","End":"03:04.770","Text":"We are going backwards actually."},{"Start":"03:04.770 ","End":"03:10.120","Text":"Each step, we\u0027re going to start from here and end here, we\u0027re working backwards."},{"Start":"03:10.120 ","End":"03:16.550","Text":"What I\u0027m going to do is write x minus 3 times 7 is less than Epsilon."},{"Start":"03:16.550 ","End":"03:18.260","Text":"Because if this is true,"},{"Start":"03:18.260 ","End":"03:23.650","Text":"then this is true because I\u0027m replacing 7 by something smaller."},{"Start":"03:23.650 ","End":"03:30.100","Text":"Now we\u0027re in a good position because we have x minus 3 times a constant."},{"Start":"03:30.100 ","End":"03:32.810","Text":"We can divide by that constant and say,"},{"Start":"03:32.810 ","End":"03:38.930","Text":"absolute value of x minus 3 is less than Epsilon over 7."},{"Start":"03:38.930 ","End":"03:42.470","Text":"We would normally take Delta to equal this"},{"Start":"03:42.470 ","End":"03:45.920","Text":"but I\u0027m not going to take Delta to be Epsilon over 7"},{"Start":"03:45.920 ","End":"03:49.415","Text":"because remember we also assume that Delta is less than 1."},{"Start":"03:49.415 ","End":"03:56.085","Text":"The trick to do is just to take the minimum of Epsilon over 7 and 1,"},{"Start":"03:56.085 ","End":"03:57.800","Text":"and now if this is true,"},{"Start":"03:57.800 ","End":"04:06.005","Text":"then we can retrace our steps and say that if x minus 3 is less than Delta,"},{"Start":"04:06.005 ","End":"04:08.369","Text":"well, and bigger than 0."},{"Start":"04:08.369 ","End":"04:11.205","Text":"We don\u0027t need the bigger than 0,"},{"Start":"04:11.205 ","End":"04:12.720","Text":"just going to ignore that."},{"Start":"04:12.720 ","End":"04:14.450","Text":"From here, if it\u0027s equal to the minimum,"},{"Start":"04:14.450 ","End":"04:17.660","Text":"then Delta is less than or equal to Epsilon over 7."},{"Start":"04:17.660 ","End":"04:20.740","Text":"From here, I get to this step,"},{"Start":"04:20.740 ","End":"04:22.980","Text":"and then I\u0027m just working my way backwards."},{"Start":"04:22.980 ","End":"04:24.545","Text":"From here to here,"},{"Start":"04:24.545 ","End":"04:29.690","Text":"and then to here to here and so we really do have that,"},{"Start":"04:29.690 ","End":"04:31.535","Text":"if this is true, then this is true."},{"Start":"04:31.535 ","End":"04:36.320","Text":"It takes some getting used to to remember that we\u0027re actually working backwards,"},{"Start":"04:36.320 ","End":"04:40.690","Text":"that this is the end result that we know, the inequality with Epsilon,"},{"Start":"04:40.690 ","End":"04:45.600","Text":"we have to figure out an inequality on Delta to make this true,"},{"Start":"04:45.600 ","End":"04:47.535","Text":"and in our case,"},{"Start":"04:47.535 ","End":"04:50.460","Text":"this is the inequality that we take."},{"Start":"04:50.460 ","End":"04:53.825","Text":"That if Delta is the minimum of Epsilon over 7 and 1,"},{"Start":"04:53.825 ","End":"04:57.050","Text":"then we\u0027re guaranteed that this inference works,"},{"Start":"04:57.050 ","End":"04:59.240","Text":"that this is true whenever this is true."},{"Start":"04:59.240 ","End":"05:02.760","Text":"Okay. We\u0027re done."}],"ID":8312},{"Watched":false,"Name":"Exercise 3","Duration":"5m 35s","ChapterTopicVideoID":8159,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this exercise, we have to use the definition of a limit to"},{"Start":"00:03.240 ","End":"00:06.645","Text":"prove that this limit holds and when we say the definition,"},{"Start":"00:06.645 ","End":"00:11.010","Text":"I\u0027m talking about the definition with the epsilon Delta."},{"Start":"00:11.010 ","End":"00:14.490","Text":"I also want to point out that sometimes"},{"Start":"00:14.490 ","End":"00:17.570","Text":"the exercises given as to find and prove."},{"Start":"00:17.570 ","End":"00:20.770","Text":"In other words, suppose I wasn\u0027t given this,"},{"Start":"00:20.770 ","End":"00:22.860","Text":"then we can easily compute the limit"},{"Start":"00:22.860 ","End":"00:24.180","Text":"to the old fashioned techniques."},{"Start":"00:24.180 ","End":"00:27.630","Text":"In this case, it\u0027s a continuous to polynomial"},{"Start":"00:27.630 ","End":"00:29.370","Text":"that we could just substitute x equals 1,"},{"Start":"00:29.370 ","End":"00:31.755","Text":"1 squared minus 1 is 0."},{"Start":"00:31.755 ","End":"00:33.630","Text":"If this limit is not given,"},{"Start":"00:33.630 ","End":"00:35.100","Text":"you can easily compute it anyway."},{"Start":"00:35.100 ","End":"00:39.945","Text":"Now we have to go about using the epsilon Delta definition."},{"Start":"00:39.945 ","End":"00:42.655","Text":"I\u0027m going to remind you,"},{"Start":"00:42.655 ","End":"00:46.280","Text":"this line here is a definition in general and just"},{"Start":"00:46.280 ","End":"00:49.340","Text":"indicated here and in our case, this is the a,"},{"Start":"00:49.340 ","End":"00:53.510","Text":"this is the f of x and this is the limit l. We"},{"Start":"00:53.510 ","End":"00:57.770","Text":"find Delta for this given epsilon such that if this is true,"},{"Start":"00:57.770 ","End":"00:59.075","Text":"then this is true."},{"Start":"00:59.075 ","End":"01:03.710","Text":"We\u0027ll start off with this and this says that"},{"Start":"01:03.710 ","End":"01:09.200","Text":"the absolute value f of x is x squared minus 1 minus 0,"},{"Start":"01:09.200 ","End":"01:13.130","Text":"won\u0027t bother with that is less than epsilon."},{"Start":"01:13.130 ","End":"01:15.440","Text":"It\u0027s the first line I write, but in a sense,"},{"Start":"01:15.440 ","End":"01:18.620","Text":"this is the end that we\u0027ll start off with something"},{"Start":"01:18.620 ","End":"01:22.830","Text":"less than Delta and end up with this logically."},{"Start":"01:23.440 ","End":"01:27.260","Text":"We factorize this because we are somehow"},{"Start":"01:27.260 ","End":"01:30.590","Text":"trying to get stuff in terms of this,"},{"Start":"01:30.590 ","End":"01:34.000","Text":"which is x minus 1, because a is 1."},{"Start":"01:34.000 ","End":"01:36.440","Text":"That\u0027s easy to do if we just break it up"},{"Start":"01:36.440 ","End":"01:38.450","Text":"using difference of squares,"},{"Start":"01:38.450 ","End":"01:43.610","Text":"x squared minus 1 squared is x minus 1 times x plus"},{"Start":"01:43.610 ","End":"01:49.685","Text":"1 and the absolute value stays in less than epsilon."},{"Start":"01:49.685 ","End":"01:53.720","Text":"Now we\u0027re going to have to use some estimations,"},{"Start":"01:53.720 ","End":"01:58.295","Text":"because if this was a constant here,"},{"Start":"01:58.295 ","End":"02:00.170","Text":"we\u0027d have no problem if this was"},{"Start":"02:00.170 ","End":"02:02.855","Text":"something like we previously had 7,"},{"Start":"02:02.855 ","End":"02:05.930","Text":"we would just divide both sides by 7 and say that x"},{"Start":"02:05.930 ","End":"02:09.530","Text":"minus 1 is less than epsilon over 7 and let that be Delta."},{"Start":"02:09.530 ","End":"02:12.920","Text":"But we don\u0027t know what this is and in general,"},{"Start":"02:12.920 ","End":"02:16.010","Text":"we have to estimate all the factors"},{"Start":"02:16.010 ","End":"02:18.350","Text":"that are not just x minus 1,"},{"Start":"02:18.350 ","End":"02:21.090","Text":"which is x minus a."},{"Start":"02:21.160 ","End":"02:27.650","Text":"What we do is a bit of algebra with absolute value and"},{"Start":"02:27.650 ","End":"02:30.350","Text":"the standard trick is to first of all assume that"},{"Start":"02:30.350 ","End":"02:34.670","Text":"absolute value of x minus 1 is less than usually 1."},{"Start":"02:34.670 ","End":"02:37.865","Text":"If 1 doesn\u0027t work for some reason, try something smaller."},{"Start":"02:37.865 ","End":"02:40.535","Text":"We can always make sure that the Delta is"},{"Start":"02:40.535 ","End":"02:43.550","Text":"less than 1, other words we assume right off"},{"Start":"02:43.550 ","End":"02:45.830","Text":"the bat that Delta is less than 1."},{"Start":"02:45.830 ","End":"02:48.185","Text":"If it isn\u0027t, we can always shrink Delta."},{"Start":"02:48.185 ","End":"02:50.420","Text":"A smaller Delta will always work too."},{"Start":"02:50.420 ","End":"02:55.405","Text":"Delta less than or equal to 1 or whatever,"},{"Start":"02:55.405 ","End":"02:58.940","Text":"will mean that x minus 1 is less than 1."},{"Start":"02:58.940 ","End":"03:00.890","Text":"If I is less than Delta,"},{"Start":"03:00.890 ","End":"03:05.020","Text":"then it\u0027s going to be less than 1."},{"Start":"03:05.020 ","End":"03:08.600","Text":"Then we solve this absolute value inequality"},{"Start":"03:08.600 ","End":"03:09.740","Text":"and I\u0027ll just tell you that it"},{"Start":"03:09.740 ","End":"03:14.285","Text":"comes out to be that x is between 0 and 2."},{"Start":"03:14.285 ","End":"03:16.160","Text":"If you think about it, it says the distance"},{"Start":"03:16.160 ","End":"03:19.800","Text":"from x to 1 is less than 1."},{"Start":"03:19.800 ","End":"03:24.315","Text":"I go 1 on either side of 1 and I get to 0 and to 2."},{"Start":"03:24.315 ","End":"03:25.980","Text":"Now what this will give me,"},{"Start":"03:25.980 ","End":"03:28.360","Text":"is it\u0027ll give me an estimate for x plus 1."},{"Start":"03:28.360 ","End":"03:32.980","Text":"X plus 1 will be between 1 and 3."},{"Start":"03:34.160 ","End":"03:37.610","Text":"Doesn\u0027t hurt to write absolute value here"},{"Start":"03:37.610 ","End":"03:38.630","Text":"because it\u0027s positive,"},{"Start":"03:38.630 ","End":"03:42.230","Text":"it\u0027s between 0 and 2 and so now we can"},{"Start":"03:42.230 ","End":"03:44.180","Text":"get an estimate for this."},{"Start":"03:44.180 ","End":"03:46.040","Text":"The question is which estimate do I want"},{"Start":"03:46.040 ","End":"03:47.390","Text":"the lower or the upper?"},{"Start":"03:47.390 ","End":"03:49.490","Text":"The answer is that I want the upper"},{"Start":"03:49.490 ","End":"03:51.670","Text":"because we\u0027re going backwards."},{"Start":"03:51.670 ","End":"03:54.200","Text":"It want something that if something is true,"},{"Start":"03:54.200 ","End":"03:55.280","Text":"then this will be true."},{"Start":"03:55.280 ","End":"03:59.270","Text":"If I replace x plus 1 by something bigger,"},{"Start":"03:59.270 ","End":"04:04.760","Text":"say 3, so I get x minus 1 times 3 is less than epsilon."},{"Start":"04:04.760 ","End":"04:07.040","Text":"If this is true, then certainly this is true,"},{"Start":"04:07.040 ","End":"04:09.725","Text":"because this is less than or equal to this,"},{"Start":"04:09.725 ","End":"04:13.370","Text":"or less than this. That\u0027s the way we do it."},{"Start":"04:13.370 ","End":"04:17.855","Text":"We go for the upper limit and now I will proceed."},{"Start":"04:17.855 ","End":"04:20.465","Text":"We say that x minus 1,"},{"Start":"04:20.465 ","End":"04:21.650","Text":"we\u0027ve done this kind of thing before,"},{"Start":"04:21.650 ","End":"04:23.960","Text":"is less than epsilon over 3."},{"Start":"04:23.960 ","End":"04:28.520","Text":"We take Delta to be not exactly"},{"Start":"04:28.520 ","End":"04:30.880","Text":"epsilon over 3 because we also assumed"},{"Start":"04:30.880 ","End":"04:33.605","Text":"that Delta less than or equal to 1."},{"Start":"04:33.605 ","End":"04:41.540","Text":"We take the minimum of epsilon 3 over 3 and 1,"},{"Start":"04:41.540 ","End":"04:43.435","Text":"the lesser of the 2."},{"Start":"04:43.435 ","End":"04:46.960","Text":"We can always push it down to make sure it\u0027s less than 1."},{"Start":"04:46.960 ","End":"04:49.910","Text":"Now this is the Delta that does the trick."},{"Start":"04:49.910 ","End":"04:53.390","Text":"Really what we would be doing would be going backwards."},{"Start":"04:53.390 ","End":"05:02.115","Text":"Now we would start from 0 less than x minus a is 1,"},{"Start":"05:02.115 ","End":"05:05.400","Text":"is less than Delta."},{"Start":"05:05.400 ","End":"05:07.569","Text":"If this is true,"},{"Start":"05:07.569 ","End":"05:09.990","Text":"and in this case I don\u0027t need this bit,"},{"Start":"05:09.990 ","End":"05:11.990","Text":"so certainly this is less than this,"},{"Start":"05:11.990 ","End":"05:15.320","Text":"but Delta is less than or equal to epsilon over 3."},{"Start":"05:15.320 ","End":"05:18.380","Text":"From here I get to this, from this line,"},{"Start":"05:18.380 ","End":"05:21.185","Text":"I get to this line which gives me to this line,"},{"Start":"05:21.185 ","End":"05:22.820","Text":"which is just this."},{"Start":"05:22.820 ","End":"05:24.380","Text":"We can work our way backwards."},{"Start":"05:24.380 ","End":"05:26.015","Text":"We\u0027ve shown that if this is true,"},{"Start":"05:26.015 ","End":"05:28.570","Text":"this is true and"},{"Start":"05:28.570 ","End":"05:30.980","Text":"sometimes I just like to highlight the formula"},{"Start":"05:30.980 ","End":"05:32.855","Text":"for Delta in terms of epsilon."},{"Start":"05:32.855 ","End":"05:36.330","Text":"This is what it is, and we\u0027re done."}],"ID":8313},{"Watched":false,"Name":"Exercise 4","Duration":"7m 57s","ChapterTopicVideoID":8160,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"In this exercise, we have to use the definition of"},{"Start":"00:03.090 ","End":"00:06.090","Text":"the limit and I\u0027m talking about the Epsilon Delta definition,"},{"Start":"00:06.090 ","End":"00:10.815","Text":"so you know what I mean, to prove that the limit of this is this."},{"Start":"00:10.815 ","End":"00:13.020","Text":"By the way, sometimes this isn\u0027t given."},{"Start":"00:13.020 ","End":"00:14.670","Text":"If I hadn\u0027t given you this,"},{"Start":"00:14.670 ","End":"00:21.660","Text":"you would have been able to deduce it because we just would have substituted x equals 24,"},{"Start":"00:21.660 ","End":"00:25.875","Text":"since this is a continuous function around x equals 24."},{"Start":"00:25.875 ","End":"00:28.200","Text":"Would have said 24 plus 1 is 25,"},{"Start":"00:28.200 ","End":"00:30.420","Text":"square root of that is 5,"},{"Start":"00:30.420 ","End":"00:32.520","Text":"and here\u0027s the 5."},{"Start":"00:32.520 ","End":"00:37.230","Text":"For the solution, I\u0027m going to remind you of the formula, not the formula,"},{"Start":"00:37.230 ","End":"00:42.980","Text":"I mean the definition which says here that if we\u0027re given Epsilon bigger than 0,"},{"Start":"00:42.980 ","End":"00:47.750","Text":"we have to find Delta such that f of x is"},{"Start":"00:47.750 ","End":"00:53.120","Text":"close to L within Epsilon whenever x is close to a within Delta,"},{"Start":"00:53.120 ","End":"00:55.010","Text":"except for x equals a."},{"Start":"00:55.010 ","End":"00:56.480","Text":"That\u0027s basically what it says."},{"Start":"00:56.480 ","End":"00:59.810","Text":"In our case we have L,"},{"Start":"00:59.810 ","End":"01:03.450","Text":"f of x and a as indicated."},{"Start":"01:04.070 ","End":"01:07.820","Text":"Even though the logical implication is"},{"Start":"01:07.820 ","End":"01:11.750","Text":"that this condition on Delta implies the condition on Epsilon,"},{"Start":"01:11.750 ","End":"01:15.229","Text":"we start from the end and we start from the Epsilon condition."},{"Start":"01:15.229 ","End":"01:16.535","Text":"If I write this out,"},{"Start":"01:16.535 ","End":"01:19.340","Text":"this says that the absolute value of f of x,"},{"Start":"01:19.340 ","End":"01:26.610","Text":"which is square root of x plus 1 minus L,"},{"Start":"01:26.610 ","End":"01:29.760","Text":"has got to be less than Epsilon."},{"Start":"01:29.760 ","End":"01:32.545","Text":"We\u0027ve got to somehow out of this,"},{"Start":"01:32.545 ","End":"01:36.140","Text":"get a condition on x minus a,"},{"Start":"01:36.140 ","End":"01:41.044","Text":"which in this case is x minus 24 somehow."},{"Start":"01:41.044 ","End":"01:43.600","Text":"Let\u0027s see what we can do."},{"Start":"01:43.600 ","End":"01:47.690","Text":"When we\u0027re dealing with square roots and plus or minus,"},{"Start":"01:47.690 ","End":"01:50.990","Text":"the thing to do is to use the conjugate."},{"Start":"01:50.990 ","End":"01:52.940","Text":"The conjugate is the same thing,"},{"Start":"01:52.940 ","End":"01:55.280","Text":"but with the opposite sign."},{"Start":"01:55.280 ","End":"02:00.845","Text":"Here I take a plus 5 and I also"},{"Start":"02:00.845 ","End":"02:06.920","Text":"multiply it by the original x plus 1 minus 5."},{"Start":"02:06.920 ","End":"02:09.740","Text":"But of course, I can\u0027t just multiply by something,"},{"Start":"02:09.740 ","End":"02:13.970","Text":"it changes the balance so I\u0027m dividing by 2."},{"Start":"02:13.970 ","End":"02:21.660","Text":"I divide by the square root of x plus 1 plus 5, absolute value."},{"Start":"02:21.660 ","End":"02:28.565","Text":"Then this over this cancels so left-hand side is unchanged and still less than Epsilon."},{"Start":"02:28.565 ","End":"02:32.344","Text":"I want to simplify the numerator at the side."},{"Start":"02:32.344 ","End":"02:38.600","Text":"If I have absolute value of a plus b times a minus b,"},{"Start":"02:38.600 ","End":"02:41.525","Text":"absolute values don\u0027t really matter there,"},{"Start":"02:41.525 ","End":"02:46.580","Text":"this would equal the absolute value of a squared minus b squared."},{"Start":"02:46.580 ","End":"02:52.400","Text":"In our case, what we would get is the absolute value of x plus"},{"Start":"02:52.400 ","End":"02:59.880","Text":"1 square root squared minus 5 squared,"},{"Start":"03:00.580 ","End":"03:07.110","Text":"this is just x plus 1 and this is 25."},{"Start":"03:07.110 ","End":"03:09.720","Text":"It\u0027s x plus 1 minus 25,"},{"Start":"03:09.720 ","End":"03:12.765","Text":"which is x minus 24."},{"Start":"03:12.765 ","End":"03:16.195","Text":"This is good. This is what I have here."},{"Start":"03:16.195 ","End":"03:24.500","Text":"What we get now is the absolute value of x minus 24 over"},{"Start":"03:24.500 ","End":"03:30.140","Text":"the absolute value of square root of x plus"},{"Start":"03:30.140 ","End":"03:36.525","Text":"1 plus 5 is less than Epsilon."},{"Start":"03:36.525 ","End":"03:39.979","Text":"Actually since this denominator is positive,"},{"Start":"03:39.979 ","End":"03:41.900","Text":"what\u0027s inside the absolute value,"},{"Start":"03:41.900 ","End":"03:45.100","Text":"I can just erase the bars."},{"Start":"03:45.100 ","End":"03:53.420","Text":"We would really like to have a constant times x minus 24 or this divided by a constant,"},{"Start":"03:53.420 ","End":"03:54.800","Text":"but this is not a constant."},{"Start":"03:54.800 ","End":"03:57.230","Text":"We want to somehow estimate this,"},{"Start":"03:57.230 ","End":"04:06.440","Text":"and our usual trick is to take Delta less than or equal to 1 usually works."},{"Start":"04:06.440 ","End":"04:10.100","Text":"If we restrict Delta to being less than or equal to 1,"},{"Start":"04:10.100 ","End":"04:12.650","Text":"it doesn\u0027t hurt because you can always shrink Delta,"},{"Start":"04:12.650 ","End":"04:19.850","Text":"then we get that absolute value of x minus 24 is less than 1."},{"Start":"04:19.850 ","End":"04:22.280","Text":"If it\u0027s less than Delta, then it\u0027s going to be less than 1."},{"Start":"04:22.280 ","End":"04:25.100","Text":"Then using absolute value inequalities,"},{"Start":"04:25.100 ","End":"04:32.310","Text":"this means that the distance from x to 24 is less than 1 so"},{"Start":"04:32.310 ","End":"04:39.480","Text":"x has to be between 25 and 23."},{"Start":"04:39.480 ","End":"04:48.600","Text":"But we want x plus 1 is going to be between 26 and 24."},{"Start":"04:48.600 ","End":"04:53.050","Text":"I can just put a square root on all of them,"},{"Start":"04:53.050 ","End":"04:54.440","Text":"so square root of this,"},{"Start":"04:54.440 ","End":"04:55.835","Text":"square root of this,"},{"Start":"04:55.835 ","End":"04:58.135","Text":"square root of this."},{"Start":"04:58.135 ","End":"05:04.140","Text":"The question is, which do we want to estimate it as the upper one of the low one?"},{"Start":"05:05.090 ","End":"05:08.420","Text":"First of all, that we\u0027re working backwards,"},{"Start":"05:08.420 ","End":"05:16.935","Text":"and I want to replace this expression by a larger expression."},{"Start":"05:16.935 ","End":"05:19.580","Text":"The whole of this are replaced by something larger,"},{"Start":"05:19.580 ","End":"05:21.380","Text":"and then if that\u0027s less than Epsilon,"},{"Start":"05:21.380 ","End":"05:24.500","Text":"something small will also be less than Epsilon."},{"Start":"05:24.500 ","End":"05:26.840","Text":"To make this fraction larger,"},{"Start":"05:26.840 ","End":"05:29.245","Text":"I want to make the denominator smaller."},{"Start":"05:29.245 ","End":"05:36.470","Text":"So I\u0027m going to say that what I want is x minus 24"},{"Start":"05:36.470 ","End":"05:45.320","Text":"over the square root of 24 plus 5 less than Epsilon."},{"Start":"05:45.320 ","End":"05:48.139","Text":"If I can get this to be true,"},{"Start":"05:48.139 ","End":"05:50.835","Text":"then this will be true,"},{"Start":"05:50.835 ","End":"05:52.954","Text":"because from here to here,"},{"Start":"05:52.954 ","End":"05:57.380","Text":"I am increasing the denominator and therefore decreasing this."},{"Start":"05:57.380 ","End":"06:01.760","Text":"If this is an Epsilon, something smaller will also be less than Epsilon."},{"Start":"06:01.760 ","End":"06:08.840","Text":"Now from here, we can say that absolute value of x minus"},{"Start":"06:08.840 ","End":"06:13.705","Text":"24 is less than"},{"Start":"06:13.705 ","End":"06:19.930","Text":"Epsilon times root 24 plus 5."},{"Start":"06:20.270 ","End":"06:26.500","Text":"If I choose this to be my Delta,"},{"Start":"06:26.500 ","End":"06:29.345","Text":"well, delta is going to be less than or equal to this."},{"Start":"06:29.345 ","End":"06:34.280","Text":"In fact, what I\u0027m going to do now is choose Delta to be the minimum,"},{"Start":"06:34.280 ","End":"06:35.630","Text":"this is our usual trick,"},{"Start":"06:35.630 ","End":"06:37.670","Text":"it has to be less than 1."},{"Start":"06:37.670 ","End":"06:39.475","Text":"We\u0027ve already used the 1 up,"},{"Start":"06:39.475 ","End":"06:47.490","Text":"the minimum between 1 and epsilon times root 24 plus 5."},{"Start":"06:47.490 ","End":"06:49.655","Text":"This will do the trick,"},{"Start":"06:49.655 ","End":"06:51.815","Text":"and I will just highlight it."},{"Start":"06:51.815 ","End":"06:53.875","Text":"Let\u0027s just quickly mention,"},{"Start":"06:53.875 ","End":"06:56.180","Text":"again the logic, if we start from this,"},{"Start":"06:56.180 ","End":"07:04.865","Text":"then it was if we start from 0 less than x minus 24, less than Delta."},{"Start":"07:04.865 ","End":"07:06.935","Text":"If this is true,"},{"Start":"07:06.935 ","End":"07:09.590","Text":"then in most cases this is what happens."},{"Start":"07:09.590 ","End":"07:13.130","Text":"I just ignore this part so this is less than this."},{"Start":"07:13.130 ","End":"07:19.575","Text":"But Delta is less than or equal to this,"},{"Start":"07:19.575 ","End":"07:29.560","Text":"so it implies that from here I can get this because Delta\u0027s less than or equal to this."},{"Start":"07:29.560 ","End":"07:31.020","Text":"If this is true,"},{"Start":"07:31.020 ","End":"07:32.550","Text":"then this is true,"},{"Start":"07:32.550 ","End":"07:33.810","Text":"then this is true,"},{"Start":"07:33.810 ","End":"07:35.100","Text":"then this is true,"},{"Start":"07:35.100 ","End":"07:36.630","Text":"then this is true."},{"Start":"07:36.630 ","End":"07:38.805","Text":"We\u0027ve gotten from here to here,"},{"Start":"07:38.805 ","End":"07:41.900","Text":"and I like to keep reminding you that we are in a sense working"},{"Start":"07:41.900 ","End":"07:47.570","Text":"backwards so that the logic has to flow from the bottom to the top."},{"Start":"07:47.570 ","End":"07:52.340","Text":"That if the Delta inequality holds then the Epsilon inequality also holds."},{"Start":"07:52.340 ","End":"07:54.695","Text":"This formula will do the trick for us,"},{"Start":"07:54.695 ","End":"07:57.330","Text":"and so we are done."}],"ID":8314},{"Watched":false,"Name":"Exercise 5","Duration":"5m 34s","ChapterTopicVideoID":8161,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this exercise, we\u0027re going to use the definition of the limit,"},{"Start":"00:04.005 ","End":"00:09.225","Text":"the Epsilon Delta definition to prove the following limit."},{"Start":"00:09.225 ","End":"00:13.140","Text":"I\u0027ll remind you what the definition is."},{"Start":"00:13.140 ","End":"00:17.360","Text":"Here it is. You\u0027ve seen it enough time so I won\u0027t go over it again."},{"Start":"00:17.360 ","End":"00:22.580","Text":"I\u0027ve copied the original exercise and I\u0027ve marked a f of x and"},{"Start":"00:22.580 ","End":"00:28.790","Text":"L in it that correspond to a f of x and L here."},{"Start":"00:28.790 ","End":"00:31.700","Text":"As usual, we start here,"},{"Start":"00:31.700 ","End":"00:33.350","Text":"even though we\u0027re working backwards,"},{"Start":"00:33.350 ","End":"00:36.290","Text":"because we have to show that this implies this,"},{"Start":"00:36.290 ","End":"00:38.120","Text":"but we start here and end up here."},{"Start":"00:38.120 ","End":"00:40.965","Text":"In our case, f of x,"},{"Start":"00:40.965 ","End":"00:44.355","Text":"which is 1 over x minus L,"},{"Start":"00:44.355 ","End":"00:45.900","Text":"which is 1,"},{"Start":"00:45.900 ","End":"00:48.685","Text":"is less than Epsilon."},{"Start":"00:48.685 ","End":"00:56.220","Text":"We have to find some conditions on absolute value of x minus 1."},{"Start":"00:56.830 ","End":"00:59.990","Text":"We have to somehow extract x minus 1."},{"Start":"00:59.990 ","End":"01:06.440","Text":"It looks like the obvious thing to do is to put a common denominator of x."},{"Start":"01:06.440 ","End":"01:12.320","Text":"Then we\u0027ll get that the absolute value of 1 minus x over x."},{"Start":"01:12.320 ","End":"01:15.170","Text":"We can also break it up like this,"},{"Start":"01:15.170 ","End":"01:18.190","Text":"is less than Epsilon."},{"Start":"01:18.390 ","End":"01:23.060","Text":"Now, x is close to 1 and so it\u0027s positive,"},{"Start":"01:23.060 ","End":"01:25.390","Text":"so we don\u0027t need the absolute value here."},{"Start":"01:25.390 ","End":"01:28.095","Text":"I can write this as just x."},{"Start":"01:28.095 ","End":"01:29.490","Text":"Also the numerator,"},{"Start":"01:29.490 ","End":"01:34.400","Text":"I\u0027ll just switch the order and write it as x minus 1 over x,"},{"Start":"01:34.400 ","End":"01:38.195","Text":"because making it minus doesn\u0027t change the absolute value."},{"Start":"01:38.195 ","End":"01:42.410","Text":"Before, what we\u0027d like to do is everything that\u0027s not the x minus 1,"},{"Start":"01:42.410 ","End":"01:45.755","Text":"all the other expressions of x we want to try and bound them."},{"Start":"01:45.755 ","End":"01:48.020","Text":"We have an x or a 1 over x,"},{"Start":"01:48.020 ","End":"01:56.020","Text":"and we\u0027d like to see if we can find upper or lower limits as necessary."},{"Start":"01:56.840 ","End":"02:00.629","Text":"Usual trick was to say,"},{"Start":"02:00.629 ","End":"02:05.960","Text":"let\u0027s have Delta less than or equal to 1."},{"Start":"02:05.960 ","End":"02:09.230","Text":"In which case we have from this,"},{"Start":"02:09.230 ","End":"02:14.300","Text":"that absolute value of x minus 1 is less than Delta,"},{"Start":"02:14.300 ","End":"02:19.320","Text":"therefore it\u0027s less than 1."},{"Start":"02:20.450 ","End":"02:24.830","Text":"Then using the properties of absolute value,"},{"Start":"02:24.830 ","End":"02:28.940","Text":"this means that the distance from x to 1 is less than 1."},{"Start":"02:28.940 ","End":"02:32.870","Text":"It gives us that x is between 0 and 2."},{"Start":"02:32.870 ","End":"02:35.220","Text":"You can check this."},{"Start":"02:35.470 ","End":"02:38.090","Text":"Now the question is,"},{"Start":"02:38.090 ","End":"02:40.190","Text":"do we want the upper or the lower bound?"},{"Start":"02:40.190 ","End":"02:44.150","Text":"But we want to replace this by something larger which means we\u0027d"},{"Start":"02:44.150 ","End":"02:49.445","Text":"like to take the lower bound on x, which is the 0."},{"Start":"02:49.445 ","End":"02:57.095","Text":"The trouble is that I can\u0027t put x equals 0 here because can\u0027t divide by 0."},{"Start":"02:57.095 ","End":"03:00.660","Text":"This doesn\u0027t work."},{"Start":"03:00.770 ","End":"03:03.800","Text":"There\u0027s more than 1 way to get around this."},{"Start":"03:03.800 ","End":"03:07.355","Text":"But the most obvious is just to simply take another Delta."},{"Start":"03:07.355 ","End":"03:12.830","Text":"I mentioned that Delta less than 1 is what we usually do,"},{"Start":"03:12.830 ","End":"03:14.930","Text":"but here it runs us into trouble."},{"Start":"03:14.930 ","End":"03:21.220","Text":"How about if we restricted Delta to be less than 1/2?"},{"Start":"03:21.220 ","End":"03:23.300","Text":"We can always make Delta smaller."},{"Start":"03:23.300 ","End":"03:24.425","Text":"That\u0027s not a problem."},{"Start":"03:24.425 ","End":"03:25.895","Text":"Let\u0027s see what this gives us."},{"Start":"03:25.895 ","End":"03:32.050","Text":"This gives us that x minus 1 is less than a 1/2."},{"Start":"03:32.050 ","End":"03:34.565","Text":"If we solve the inequality,"},{"Start":"03:34.565 ","End":"03:40.470","Text":"it gives us that x is between 1/2 and 1 and 1/2."},{"Start":"03:40.470 ","End":"03:46.325","Text":"Now I\u0027m okay with taking the lower limit for x."},{"Start":"03:46.325 ","End":"03:51.560","Text":"What I can say is that if x minus"},{"Start":"03:51.560 ","End":"03:57.175","Text":"1 over this 1/2 is less than Epsilon,"},{"Start":"03:57.175 ","End":"03:59.825","Text":"then this would imply this."},{"Start":"03:59.825 ","End":"04:02.750","Text":"Because if I shrink the denominator,"},{"Start":"04:02.750 ","End":"04:08.010","Text":"I\u0027ve increased this expression."},{"Start":"04:08.010 ","End":"04:12.090","Text":"If this is bigger than this and if this is less than Epsilon,"},{"Start":"04:12.090 ","End":"04:14.105","Text":"certainly this is less than epsilon."},{"Start":"04:14.105 ","End":"04:17.310","Text":"Remember, we\u0027re working our way backwards."},{"Start":"04:17.560 ","End":"04:22.820","Text":"This will give us that if we just bring the 1/2 to the other side,"},{"Start":"04:22.820 ","End":"04:25.715","Text":"that absolute value of x minus 1,"},{"Start":"04:25.715 ","End":"04:28.715","Text":"less than a 1/2 Epsilon."},{"Start":"04:28.715 ","End":"04:34.195","Text":"That\u0027s near the end because now we can take Delta is equal to,"},{"Start":"04:34.195 ","End":"04:39.640","Text":"well, that a 1/2 Epsilon because we also had the other condition with the 1/2."},{"Start":"04:39.640 ","End":"04:44.250","Text":"I\u0027ll take the minimum of 1/2 and 1/2 Epsilon."},{"Start":"04:44.250 ","End":"04:47.375","Text":"Now we\u0027re okay because if we start from here,"},{"Start":"04:47.375 ","End":"04:54.985","Text":"which says that 0 less than absolute value of x minus 1 less than Delta."},{"Start":"04:54.985 ","End":"04:57.165","Text":"I can ignore this bit."},{"Start":"04:57.165 ","End":"05:00.675","Text":"This less than Delta implies that it\u0027s less than"},{"Start":"05:00.675 ","End":"05:04.875","Text":"a 1/2 Epsilon because Delta is less than or equal to 1/2 Epsilon."},{"Start":"05:04.875 ","End":"05:07.755","Text":"This is equivalent to this."},{"Start":"05:07.755 ","End":"05:11.030","Text":"This from here to here we already had based"},{"Start":"05:11.030 ","End":"05:14.950","Text":"on the fact that Delta is less than or equal to a 1/2."},{"Start":"05:14.950 ","End":"05:20.180","Text":"Up the chain, up the chain till we get to here, which is this."},{"Start":"05:20.180 ","End":"05:24.590","Text":"I\u0027ll just highlight the important bit which is"},{"Start":"05:24.590 ","End":"05:29.710","Text":"that Delta is minimum of 1/2 and 1/2 Epsilon."},{"Start":"05:29.710 ","End":"05:35.300","Text":"It gives us Delta in terms of Epsilon. We\u0027re done."}],"ID":8315},{"Watched":false,"Name":"Exercise 6","Duration":"5m 44s","ChapterTopicVideoID":8162,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.355","Text":"In this exercise, you have to use the definition of the limit to prove that."},{"Start":"00:05.355 ","End":"00:09.180","Text":"I just made it a bit harder, I erased this."},{"Start":"00:09.180 ","End":"00:11.565","Text":"What do you think this limit would be?"},{"Start":"00:11.565 ","End":"00:15.690","Text":"Well, sine x is a nicely behaved function, it\u0027s continuous."},{"Start":"00:15.690 ","End":"00:19.920","Text":"It should be that if we just replace x by pi over 4,"},{"Start":"00:19.920 ","End":"00:23.099","Text":"we substitute, we\u0027d get the answer."},{"Start":"00:23.099 ","End":"00:25.160","Text":"This is what we have to prove,"},{"Start":"00:25.160 ","End":"00:27.170","Text":"what is intuitively obvious,"},{"Start":"00:27.170 ","End":"00:31.955","Text":"and I\u0027m talking about the epsilon-delta definition."},{"Start":"00:31.955 ","End":"00:37.100","Text":"Here I reminded you of what that definition is in"},{"Start":"00:37.100 ","End":"00:44.690","Text":"general terms and in our case I emphasized what are a and L and f of x."},{"Start":"00:44.690 ","End":"00:48.750","Text":"We\u0027re all ready to use this."},{"Start":"00:48.750 ","End":"00:50.720","Text":"We do things in reverse."},{"Start":"00:50.720 ","End":"00:55.295","Text":"The definition says that this implies this,"},{"Start":"00:55.295 ","End":"00:59.920","Text":"but we actually start from here and work our way backwards."},{"Start":"00:59.920 ","End":"01:03.665","Text":"Let\u0027s see what is this inequality."},{"Start":"01:03.665 ","End":"01:10.010","Text":"F of x is sine x and L is"},{"Start":"01:10.010 ","End":"01:16.850","Text":"sine of 1/4 pi less than epsilon."},{"Start":"01:16.850 ","End":"01:23.404","Text":"We want this to be true provided that delta is suitably chosen."},{"Start":"01:23.404 ","End":"01:28.340","Text":"Now, how are we going to get from here to this expression,"},{"Start":"01:28.340 ","End":"01:31.060","Text":"which is x minus pi over 4?"},{"Start":"01:31.060 ","End":"01:32.870","Text":"It\u0027s not clear."},{"Start":"01:32.870 ","End":"01:35.690","Text":"We can\u0027t just take out the sine."},{"Start":"01:35.690 ","End":"01:40.715","Text":"There are several ways to do this using trigonometric identities."},{"Start":"01:40.715 ","End":"01:43.590","Text":"I want to present you with one way."},{"Start":"01:43.590 ","End":"01:47.400","Text":"But I\u0027m going to need a proposition,"},{"Start":"01:47.400 ","End":"01:49.530","Text":"and that is that."},{"Start":"01:49.530 ","End":"01:58.590","Text":"In general, sine of Alpha minus sine of Beta,"},{"Start":"01:58.590 ","End":"02:05.320","Text":"the difference in absolute value is less than or equal to Alpha minus Beta."},{"Start":"02:05.320 ","End":"02:07.550","Text":"Then we can apply that here."},{"Start":"02:07.550 ","End":"02:11.674","Text":"I hope you remember something called the intermediate value theorem."},{"Start":"02:11.674 ","End":"02:14.345","Text":"But I\u0027ll remind you of what it is."},{"Start":"02:14.345 ","End":"02:20.030","Text":"In our case, let\u0027s just assume that Alpha\u0027s less than or equal to Beta."},{"Start":"02:20.030 ","End":"02:22.160","Text":"If not, we\u0027ll just switch the roles."},{"Start":"02:22.160 ","End":"02:25.250","Text":"Now, if we take the function f of x,"},{"Start":"02:25.250 ","End":"02:27.890","Text":"which is sine x,"},{"Start":"02:27.890 ","End":"02:35.630","Text":"we have f of x defined on the closed interval from Alpha to Beta."},{"Start":"02:35.630 ","End":"02:41.870","Text":"Notice that f of x is differentiable and in"},{"Start":"02:41.870 ","End":"02:48.900","Text":"that case it guarantees that there is a point c. I switched from Alpha-Beta to a,"},{"Start":"02:48.900 ","End":"02:50.405","Text":"b, or more familiar."},{"Start":"02:50.405 ","End":"02:54.555","Text":"There\u0027s a point c between a and b."},{"Start":"02:54.555 ","End":"02:58.115","Text":"We have some intermediate value,"},{"Start":"02:58.115 ","End":"03:04.640","Text":"such that f of b minus f of"},{"Start":"03:04.640 ","End":"03:13.190","Text":"a over b minus a is f prime of c. That\u0027s the intermediate value theorem,"},{"Start":"03:13.190 ","End":"03:17.120","Text":"and what it says in our case is that"},{"Start":"03:17.120 ","End":"03:24.900","Text":"the sine x minus sine pi over 4,"},{"Start":"03:24.900 ","End":"03:32.209","Text":"divided by x minus pi over 4 is f prime."},{"Start":"03:32.209 ","End":"03:34.490","Text":"If f of x is sine x,"},{"Start":"03:34.490 ","End":"03:38.000","Text":"then f prime is cosine x."},{"Start":"03:38.000 ","End":"03:43.400","Text":"We have cosine of c. Now we can"},{"Start":"03:43.400 ","End":"03:51.140","Text":"also take the absolute value of each of these."},{"Start":"03:51.140 ","End":"03:58.355","Text":"The absolute value of the cosine is always less than or equal to 1."},{"Start":"03:58.355 ","End":"04:01.100","Text":"Cosine is between minus 1 and 1,"},{"Start":"04:01.100 ","End":"04:04.265","Text":"so it\u0027s absolute value is less than or equal to 1."},{"Start":"04:04.265 ","End":"04:10.700","Text":"What this gives us is that sine x minus"},{"Start":"04:10.700 ","End":"04:19.355","Text":"sine pi over 4 is less than or equal to x minus pi over 4,"},{"Start":"04:19.355 ","End":"04:21.560","Text":"each of these in absolute value."},{"Start":"04:21.560 ","End":"04:25.385","Text":"Now, if I have this,"},{"Start":"04:25.385 ","End":"04:29.285","Text":"then if I want to get to this inequality,"},{"Start":"04:29.285 ","End":"04:37.760","Text":"all I have to do is show that x minus pi over 4 is less than epsilon."},{"Start":"04:37.760 ","End":"04:41.720","Text":"If I can get this to be true,"},{"Start":"04:41.720 ","End":"04:44.405","Text":"then this will imply this,"},{"Start":"04:44.405 ","End":"04:48.260","Text":"because this is less than or equal to this."},{"Start":"04:48.260 ","End":"04:50.045","Text":"All I have to do now,"},{"Start":"04:50.045 ","End":"04:51.545","Text":"if we look at this,"},{"Start":"04:51.545 ","End":"04:56.040","Text":"is if I choose my Delta to be epsilon,"},{"Start":"04:57.470 ","End":"05:03.835","Text":"then we know that this is less than Delta,"},{"Start":"05:03.835 ","End":"05:07.940","Text":"x minus pi over 4 is less than Delta."},{"Start":"05:07.940 ","End":"05:09.770","Text":"It\u0027s also bigger than 0."},{"Start":"05:09.770 ","End":"05:12.590","Text":"But I don\u0027t need this part."},{"Start":"05:12.590 ","End":"05:17.480","Text":"If this is less than Delta and Delta is equal to epsilon,"},{"Start":"05:17.480 ","End":"05:20.075","Text":"then this will be true,"},{"Start":"05:20.075 ","End":"05:21.815","Text":"and hence this will be true."},{"Start":"05:21.815 ","End":"05:27.440","Text":"Of course, we had to know to use the intermediate value theorem or"},{"Start":"05:27.440 ","End":"05:33.879","Text":"perhaps it might even be listed in a formula sheet that this is true."},{"Start":"05:33.879 ","End":"05:40.160","Text":"Then from here, we have that this implies this,"},{"Start":"05:40.160 ","End":"05:44.550","Text":"which is what this says here. We\u0027re done."}],"ID":8316},{"Watched":false,"Name":"Exercise 7","Duration":"5m 47s","ChapterTopicVideoID":8163,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this exercise, we have to prove this limit"},{"Start":"00:03.900 ","End":"00:05.820","Text":"using the definition of the limit."},{"Start":"00:05.820 ","End":"00:09.975","Text":"I\u0027m talking about the Epsilon Delta definition."},{"Start":"00:09.975 ","End":"00:12.330","Text":"Let me remind you of it."},{"Start":"00:12.330 ","End":"00:17.880","Text":"Definition refers to a general case where we have a here,"},{"Start":"00:17.880 ","End":"00:21.540","Text":"f of x here, and L here and this is how it\u0027s defined."},{"Start":"00:21.540 ","End":"00:23.430","Text":"For each epsilon, we have to find Delta"},{"Start":"00:23.430 ","End":"00:25.860","Text":"such that Delta. In general,"},{"Start":"00:25.860 ","End":"00:30.560","Text":"the condition looks like that if x satisfies"},{"Start":"00:30.560 ","End":"00:33.215","Text":"some inequality in terms of Delta,"},{"Start":"00:33.215 ","End":"00:35.360","Text":"then f of x satisfies some other"},{"Start":"00:35.360 ","End":"00:37.955","Text":"inequality in terms of Epsilon."},{"Start":"00:37.955 ","End":"00:42.830","Text":"Whenever means that this implies this but in practice,"},{"Start":"00:42.830 ","End":"00:46.870","Text":"we start from this and work our way backwards to this."},{"Start":"00:46.870 ","End":"00:51.170","Text":"Here\u0027s the f of x minus L less than epsilon."},{"Start":"00:51.170 ","End":"00:55.610","Text":"We just replaced f of x and this is L. We want"},{"Start":"00:55.610 ","End":"00:59.810","Text":"to develop this simplify it in such a way"},{"Start":"00:59.810 ","End":"01:02.090","Text":"that we can see the x minus A in it."},{"Start":"01:02.090 ","End":"01:06.020","Text":"In other words, we want to see x minus 2"},{"Start":"01:06.020 ","End":"01:08.270","Text":"times something from here."},{"Start":"01:08.270 ","End":"01:10.625","Text":"Let\u0027s see what we can do."},{"Start":"01:10.625 ","End":"01:14.090","Text":"Common denominator inside the absolute value."},{"Start":"01:14.090 ","End":"01:15.890","Text":"It\u0027s all over x squared plus 1 here,"},{"Start":"01:15.890 ","End":"01:17.135","Text":"3 plus x,"},{"Start":"01:17.135 ","End":"01:20.120","Text":"and here I have to multiply by x squared plus 1."},{"Start":"01:20.120 ","End":"01:21.485","Text":"This is what I get."},{"Start":"01:21.485 ","End":"01:24.754","Text":"Now we want to cancel some terms"},{"Start":"01:24.754 ","End":"01:28.490","Text":"and collecting together and rearranging"},{"Start":"01:28.490 ","End":"01:30.095","Text":"the order, we get this."},{"Start":"01:30.095 ","End":"01:34.470","Text":"Now the numerator I claim factorizes."},{"Start":"01:35.210 ","End":"01:38.360","Text":"I\u0027m just going to give you a result of the factorization"},{"Start":"01:38.360 ","End":"01:39.800","Text":"that took the minus out"},{"Start":"01:39.800 ","End":"01:43.970","Text":"and then we had x squared minus x minus 2."},{"Start":"01:43.970 ","End":"01:46.160","Text":"If you solve it, you get 2 solutions,"},{"Start":"01:46.160 ","End":"01:47.795","Text":"2 and minus 1."},{"Start":"01:47.795 ","End":"01:50.500","Text":"Anyway, this is what we get."},{"Start":"01:50.500 ","End":"01:53.540","Text":"Now I see the x minus 2 I was looking"},{"Start":"01:53.540 ","End":"01:58.520","Text":"for earlier so I split that off."},{"Start":"01:58.520 ","End":"02:01.440","Text":"Also, I can throw out the minus because it\u0027s absolute value."},{"Start":"02:01.440 ","End":"02:05.915","Text":"I\u0027ve got x minus 2 times this over this in absolute value."},{"Start":"02:05.915 ","End":"02:08.210","Text":"Also, I don\u0027t need absolute value here"},{"Start":"02:08.210 ","End":"02:11.305","Text":"because x squared plus 1 is positive."},{"Start":"02:11.305 ","End":"02:13.725","Text":"We are at this point."},{"Start":"02:13.725 ","End":"02:18.110","Text":"What we have to do here is to try and evaluate"},{"Start":"02:18.110 ","End":"02:20.150","Text":"this as being less than some"},{"Start":"02:20.150 ","End":"02:23.900","Text":"constant and then we know how to deal"},{"Start":"02:23.900 ","End":"02:27.230","Text":"with that because we can divide by the constant."},{"Start":"02:27.230 ","End":"02:30.020","Text":"Let\u0027s see what we can do."},{"Start":"02:30.020 ","End":"02:33.715","Text":"We\u0027re going to do our usual trick in making estimates."},{"Start":"02:33.715 ","End":"02:35.710","Text":"We know that ultimately we are going"},{"Start":"02:35.710 ","End":"02:37.300","Text":"to have x minus 2 less than"},{"Start":"02:37.300 ","End":"02:40.884","Text":"Delta and we can always choose Delta small as we please."},{"Start":"02:40.884 ","End":"02:43.975","Text":"We take Delta to be less than or equal to 1,"},{"Start":"02:43.975 ","End":"02:45.355","Text":"and we make sure that,"},{"Start":"02:45.355 ","End":"02:50.370","Text":"and then we\u0027ve got that x minus 2 is less than 1."},{"Start":"02:50.370 ","End":"02:52.690","Text":"Sometimes 1 doesn\u0027t work and we have"},{"Start":"02:52.690 ","End":"02:54.535","Text":"to make another choice,"},{"Start":"02:54.535 ","End":"02:57.850","Text":"perhaps something smaller, but here it will work."},{"Start":"02:57.850 ","End":"03:00.175","Text":"If this is true,"},{"Start":"03:00.175 ","End":"03:03.430","Text":"then this inequality implies this inequality"},{"Start":"03:03.430 ","End":"03:06.380","Text":"just using properties of absolute value."},{"Start":"03:06.380 ","End":"03:11.020","Text":"This now gives us the range of possibilities for x."},{"Start":"03:11.020 ","End":"03:16.145","Text":"Now what I\u0027m interested are in these 2 pieces,"},{"Start":"03:16.145 ","End":"03:19.715","Text":"I want to estimate the x plus 1 piece."},{"Start":"03:19.715 ","End":"03:23.990","Text":"I\u0027m also interested in the x squared plus 1 factor."},{"Start":"03:23.990 ","End":"03:26.630","Text":"In general, I want to replace this fraction"},{"Start":"03:26.630 ","End":"03:28.070","Text":"by something bigger."},{"Start":"03:28.070 ","End":"03:31.055","Text":"What I want to do is make this bigger,"},{"Start":"03:31.055 ","End":"03:32.720","Text":"I less than something,"},{"Start":"03:32.720 ","End":"03:36.395","Text":"and I want to make this smaller, bigger than something."},{"Start":"03:36.395 ","End":"03:40.385","Text":"Now, we have that x is between 1 and 3."},{"Start":"03:40.385 ","End":"03:43.730","Text":"If I add 1, I\u0027ve got that x plus 1 is"},{"Start":"03:43.730 ","End":"03:48.110","Text":"between 2 and 4 but since we\u0027re in positive numbers,"},{"Start":"03:48.110 ","End":"03:50.900","Text":"then we can put absolute value."},{"Start":"03:50.900 ","End":"03:53.900","Text":"That\u0027s the numerator."},{"Start":"03:53.900 ","End":"03:56.420","Text":"What\u0027s important to me that it\u0027s less than 4"},{"Start":"03:56.420 ","End":"03:58.580","Text":"the let bigger than 2 I don\u0027t care about."},{"Start":"03:58.580 ","End":"03:59.990","Text":"With the other 1,"},{"Start":"03:59.990 ","End":"04:01.670","Text":"this gives me this, well,"},{"Start":"04:01.670 ","End":"04:03.650","Text":"I can put an intermediate step, we square it,"},{"Start":"04:03.650 ","End":"04:07.010","Text":"then we get that 1 is less than x squared,"},{"Start":"04:07.010 ","End":"04:09.455","Text":"less than 9, and then add another 1."},{"Start":"04:09.455 ","End":"04:12.685","Text":"We\u0027ve got this, and this time,"},{"Start":"04:12.685 ","End":"04:15.140","Text":"I\u0027ve got this between 2 and 10."},{"Start":"04:15.140 ","End":"04:16.490","Text":"I want the lower limit,"},{"Start":"04:16.490 ","End":"04:18.890","Text":"and that\u0027s why I\u0027ve highlighted or colored"},{"Start":"04:18.890 ","End":"04:21.065","Text":"this in a different color."},{"Start":"04:21.065 ","End":"04:27.210","Text":"Now I can replace these 2 and just a second."},{"Start":"04:29.020 ","End":"04:31.990","Text":"Hereby the 4 which is bigger and here"},{"Start":"04:31.990 ","End":"04:33.750","Text":"by the 2 which is smaller."},{"Start":"04:33.750 ","End":"04:36.570","Text":"This fraction is less than 2."},{"Start":"04:36.570 ","End":"04:41.235","Text":"What I\u0027m going to do is replace this by 2."},{"Start":"04:41.235 ","End":"04:44.750","Text":"Then if the larger expression is less than Epsilon,"},{"Start":"04:44.750 ","End":"04:49.700","Text":"then this 1 will also be less than Epsilon so just a second."},{"Start":"04:49.710 ","End":"04:55.520","Text":"Our expression is less than 2, I put the 2 in front"},{"Start":"04:55.520 ","End":"04:57.830","Text":"times absolute value of x minus 2."},{"Start":"04:57.830 ","End":"04:59.270","Text":"But this is less than Delta,"},{"Start":"04:59.270 ","End":"05:01.310","Text":"so we have less than 2 Delta."},{"Start":"05:01.310 ","End":"05:05.210","Text":"It makes sense to choose Delta less"},{"Start":"05:05.210 ","End":"05:08.350","Text":"than or less than or equal to epsilon over 2"},{"Start":"05:08.350 ","End":"05:09.775","Text":"because if that\u0027s the case,"},{"Start":"05:09.775 ","End":"05:15.695","Text":"then the 2 Delta will be less than Epsilon."},{"Start":"05:15.695 ","End":"05:20.745","Text":"We chose that Delta to be smaller or equal to 1."},{"Start":"05:20.745 ","End":"05:22.100","Text":"Here we have another condition,"},{"Start":"05:22.100 ","End":"05:23.900","Text":"smaller or equal to Epsilon over 2."},{"Start":"05:23.900 ","End":"05:26.045","Text":"If I want both of these to hold,"},{"Start":"05:26.045 ","End":"05:30.385","Text":"then I just take the minimum of these 2."},{"Start":"05:30.385 ","End":"05:36.315","Text":"If Delta is the minimum between 1 epsilon over 2,"},{"Start":"05:36.315 ","End":"05:39.900","Text":"then all this work goes backwards"},{"Start":"05:39.900 ","End":"05:44.210","Text":"and we\u0027ll get the result that we wanted."},{"Start":"05:44.210 ","End":"05:46.325","Text":"I won\u0027t go through it backwards again."},{"Start":"05:46.325 ","End":"05:48.780","Text":"We are now done."}],"ID":8317},{"Watched":false,"Name":"Exercise 8","Duration":"4m 33s","ChapterTopicVideoID":8164,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we\u0027re going to use the definition of the limit."},{"Start":"00:04.320 ","End":"00:08.460","Text":"We\u0027re talking about the Epsilon Delta definition"},{"Start":"00:08.460 ","End":"00:10.680","Text":"to prove the following limit."},{"Start":"00:10.680 ","End":"00:14.535","Text":"The limit as x goes to 4 from below,"},{"Start":"00:14.535 ","End":"00:17.400","Text":"notice that there\u0027s a little minus here"},{"Start":"00:17.400 ","End":"00:19.890","Text":"of square root of 4 minus x equals 0."},{"Start":"00:19.890 ","End":"00:22.020","Text":"Notice that it makes sense to take"},{"Start":"00:22.020 ","End":"00:24.285","Text":"the limit from below because"},{"Start":"00:24.285 ","End":"00:28.470","Text":"this square root is only defined when"},{"Start":"00:28.470 ","End":"00:30.810","Text":"the argument is non-negative,"},{"Start":"00:30.810 ","End":"00:34.880","Text":"which means that x has to be less than"},{"Start":"00:34.880 ","End":"00:37.305","Text":"or equal to 4 for the domain."},{"Start":"00:37.305 ","End":"00:40.105","Text":"That\u0027s why we can only take a limit from below."},{"Start":"00:40.105 ","End":"00:45.395","Text":"They brought the definition of the limit from the left."},{"Start":"00:45.395 ","End":"00:50.180","Text":"In the general case where we have here a,"},{"Start":"00:50.180 ","End":"00:54.800","Text":"here f of x, and here l. This gives us the definition"},{"Start":"00:54.800 ","End":"00:56.210","Text":"of limit as x goes to a"},{"Start":"00:56.210 ","End":"01:02.930","Text":"from the left that we\u0027re given Epsilon bigger than 0,"},{"Start":"01:02.930 ","End":"01:07.200","Text":"we have to find Delta such that and so on and so on."},{"Start":"01:07.330 ","End":"01:11.150","Text":"We have basically that this has to be true"},{"Start":"01:11.150 ","End":"01:12.290","Text":"whenever this is true."},{"Start":"01:12.290 ","End":"01:15.875","Text":"The condition is that this implies this."},{"Start":"01:15.875 ","End":"01:18.425","Text":"But when we do it in practice,"},{"Start":"01:18.425 ","End":"01:21.890","Text":"we start from here and work our way backwards to this."},{"Start":"01:21.890 ","End":"01:27.140","Text":"Now this first inequality with the Epsilon can be written"},{"Start":"01:27.140 ","End":"01:35.195","Text":"as f of x is square root of 4 minus x."},{"Start":"01:35.195 ","End":"01:39.470","Text":"Now, I have to take away l. Let\u0027s minus"},{"Start":"01:39.470 ","End":"01:44.210","Text":"0 and put it in absolute values and that\u0027s less than Epsilon."},{"Start":"01:44.210 ","End":"01:48.020","Text":"Obviously, I can throw out the 0"},{"Start":"01:48.020 ","End":"01:50.420","Text":"and the square root is a positive or"},{"Start":"01:50.420 ","End":"01:52.040","Text":"at least non-negative quantity."},{"Start":"01:52.040 ","End":"01:54.620","Text":"This just says the square root"},{"Start":"01:54.620 ","End":"01:59.785","Text":"of 4 minus x less than Epsilon."},{"Start":"01:59.785 ","End":"02:03.745","Text":"Before I continue, let\u0027s just see what I\u0027m aiming for."},{"Start":"02:03.745 ","End":"02:11.540","Text":"What we\u0027re aiming for is x minus a is x minus 4,"},{"Start":"02:11.660 ","End":"02:16.365","Text":"less than 0, greater than minus Delta."},{"Start":"02:16.365 ","End":"02:18.790","Text":"We want to find a Delta such that if this is true,"},{"Start":"02:18.790 ","End":"02:20.215","Text":"then this is true."},{"Start":"02:20.215 ","End":"02:22.015","Text":"I\u0027m going to rewrite this."},{"Start":"02:22.015 ","End":"02:25.390","Text":"Let\u0027s just multiply everything by minus 1"},{"Start":"02:25.390 ","End":"02:27.010","Text":"and that will reverse everything."},{"Start":"02:27.010 ","End":"02:31.690","Text":"This will say that 0 is less than minus"},{"Start":"02:31.690 ","End":"02:36.580","Text":"of this is 4 minus x less than Delta."},{"Start":"02:36.580 ","End":"02:39.670","Text":"The 4 minus x suits me better"},{"Start":"02:39.670 ","End":"02:41.985","Text":"because I have a 4 minus x here."},{"Start":"02:41.985 ","End":"02:45.140","Text":"Now let\u0027s square both sides."},{"Start":"02:45.140 ","End":"02:47.240","Text":"If this is less than this,"},{"Start":"02:47.240 ","End":"02:52.820","Text":"then 4 minus x is less than Epsilon squared."},{"Start":"02:52.820 ","End":"02:57.440","Text":"Now we want the implication to work backwards."},{"Start":"02:57.440 ","End":"03:00.140","Text":"It will work backwards as long as"},{"Start":"03:00.140 ","End":"03:02.165","Text":"we can take the square root to this."},{"Start":"03:02.165 ","End":"03:10.535","Text":"If we also had that 4 minus x was non-negative,"},{"Start":"03:10.535 ","End":"03:13.440","Text":"bigger or equal to 0."},{"Start":"03:13.510 ","End":"03:16.820","Text":"As well as this, then we could go back."},{"Start":"03:16.820 ","End":"03:20.810","Text":"This is not a problem because I\u0027m looking here,"},{"Start":"03:20.810 ","End":"03:26.845","Text":"and I see that if we just let Epsilon squared be Delta,"},{"Start":"03:26.845 ","End":"03:30.160","Text":"so Epsilon squared is Delta."},{"Start":"03:30.160 ","End":"03:31.550","Text":"Let me write it the other way around."},{"Start":"03:31.550 ","End":"03:34.520","Text":"Yeah, let\u0027s take Delta equals Epsilon squared."},{"Start":"03:34.520 ","End":"03:37.580","Text":"This should do the trick because now"},{"Start":"03:37.580 ","End":"03:43.220","Text":"if 0 less than 4 minus x less than Delta,"},{"Start":"03:43.220 ","End":"03:46.610","Text":"then 4 minus x would also be less than Epsilon"},{"Start":"03:46.610 ","End":"03:49.730","Text":"squared because this is equal to this."},{"Start":"03:49.730 ","End":"03:52.430","Text":"It can get from here to here."},{"Start":"03:52.430 ","End":"03:58.790","Text":"Also I have this because if this 4 minus x is bigger than 0,"},{"Start":"03:58.790 ","End":"04:01.220","Text":"in particular, bigger or equal to 0."},{"Start":"04:01.220 ","End":"04:04.490","Text":"All that we cared about that we can take the square root."},{"Start":"04:04.490 ","End":"04:07.625","Text":"We can go back to here,"},{"Start":"04:07.625 ","End":"04:09.365","Text":"and then from here,"},{"Start":"04:09.365 ","End":"04:12.125","Text":"it\u0027s the same thing as this to here."},{"Start":"04:12.125 ","End":"04:18.620","Text":"When we have the inequality on Delta holding,"},{"Start":"04:18.620 ","End":"04:21.649","Text":"then we also get this inequality on Epsilon."},{"Start":"04:21.649 ","End":"04:26.090","Text":"This is what we had to show for left-handed limit."},{"Start":"04:26.090 ","End":"04:28.610","Text":"This is the essence of it is that we"},{"Start":"04:28.610 ","End":"04:31.715","Text":"take Delta to be Epsilon squared."},{"Start":"04:31.715 ","End":"04:34.500","Text":"I\u0027m done with this 1."}],"ID":8318},{"Watched":false,"Name":"Exercise 9","Duration":"4m 46s","ChapterTopicVideoID":8165,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this exercise, you\u0027ve got to use the definition of the limit."},{"Start":"00:03.660 ","End":"00:05.430","Text":"I\u0027m talking about the Epsilon Delta"},{"Start":"00:05.430 ","End":"00:08.100","Text":"definition to prove 2 things."},{"Start":"00:08.100 ","End":"00:10.260","Text":"They look almost the same,"},{"Start":"00:10.260 ","End":"00:13.140","Text":"but notice that here I have a limit"},{"Start":"00:13.140 ","End":"00:15.180","Text":"from the right or from above,"},{"Start":"00:15.180 ","End":"00:18.690","Text":"and here I have a limit from the left or below."},{"Start":"00:18.690 ","End":"00:22.440","Text":"In both cases, it\u0027s the same function,"},{"Start":"00:22.440 ","End":"00:28.245","Text":"f of x, which is equal to absolute value of x over x."},{"Start":"00:28.245 ","End":"00:33.450","Text":"In fact, even the same point 0 except ones"},{"Start":"00:33.450 ","End":"00:37.854","Text":"from the right and ones from the left."},{"Start":"00:37.854 ","End":"00:44.285","Text":"Note also that this is actually not defined at x equals naught,"},{"Start":"00:44.285 ","End":"00:46.220","Text":"but we don\u0027t care because when we take"},{"Start":"00:46.220 ","End":"00:48.740","Text":"a limit to some value,"},{"Start":"00:48.740 ","End":"00:51.890","Text":"say 0, we don\u0027t care if the function itself"},{"Start":"00:51.890 ","End":"00:54.215","Text":"is defined at 0 or not."},{"Start":"00:54.215 ","End":"00:58.660","Text":"Let me bring in a sketch of this function."},{"Start":"00:58.660 ","End":"01:01.520","Text":"Here\u0027s the sketch. Now why does it look like this?"},{"Start":"01:01.520 ","End":"01:04.199","Text":"Let\u0027s just call this y."},{"Start":"01:04.570 ","End":"01:07.865","Text":"If x is positive,"},{"Start":"01:07.865 ","End":"01:10.910","Text":"then absolute value of x is the same as x,"},{"Start":"01:10.910 ","End":"01:13.730","Text":"so we have x over x is 1."},{"Start":"01:13.730 ","End":"01:15.950","Text":"If x is negative,"},{"Start":"01:15.950 ","End":"01:18.305","Text":"absolute value of x is minus x,"},{"Start":"01:18.305 ","End":"01:20.660","Text":"and that\u0027s why we get minus 1."},{"Start":"01:20.660 ","End":"01:23.480","Text":"We\u0027ve got minus 1 here, plus 1 here,"},{"Start":"01:23.480 ","End":"01:24.890","Text":"at 0 itself,"},{"Start":"01:24.890 ","End":"01:27.819","Text":"we don\u0027t know, it\u0027s not defined."},{"Start":"01:27.819 ","End":"01:32.715","Text":"I\u0027m going to remind you of the Epsilon Delta definitions."},{"Start":"01:32.715 ","End":"01:34.970","Text":"Here I just labeled stuff."},{"Start":"01:34.970 ","End":"01:38.180","Text":"The limit is what we call L. Here it\u0027s 1,"},{"Start":"01:38.180 ","End":"01:39.350","Text":"here it\u0027s minus 1."},{"Start":"01:39.350 ","End":"01:45.420","Text":"The function is f and what x tends to is a."},{"Start":"01:45.890 ","End":"01:49.340","Text":"Here I brought you 2 definitions in 1"},{"Start":"01:49.340 ","End":"01:51.575","Text":"for the limit from the left and the right."},{"Start":"01:51.575 ","End":"01:54.080","Text":"In both cases, we start with Epsilon"},{"Start":"01:54.080 ","End":"01:56.495","Text":"bigger than 0 given to us,"},{"Start":"01:56.495 ","End":"02:01.490","Text":"and we have to find Delta such that f of x is"},{"Start":"02:01.490 ","End":"02:04.370","Text":"close to L within Epsilon whenever,"},{"Start":"02:04.370 ","End":"02:05.945","Text":"and here\u0027s the difference."},{"Start":"02:05.945 ","End":"02:08.765","Text":"In the case of the limit from the right,"},{"Start":"02:08.765 ","End":"02:10.475","Text":"we have this condition,"},{"Start":"02:10.475 ","End":"02:11.690","Text":"on the limit from the left,"},{"Start":"02:11.690 ","End":"02:13.295","Text":"we have this condition."},{"Start":"02:13.295 ","End":"02:15.005","Text":"Let me rewrite them."},{"Start":"02:15.005 ","End":"02:20.705","Text":"We know that a in our case is 0, so if a is 0,"},{"Start":"02:20.705 ","End":"02:28.410","Text":"we can write the first 1 as x between 0 and Delta."},{"Start":"02:28.410 ","End":"02:32.930","Text":"So x is just a bit to the right of 0 up to Delta,"},{"Start":"02:32.930 ","End":"02:36.035","Text":"that\u0027s the right-handed limit, and in the left-handed limit,"},{"Start":"02:36.035 ","End":"02:41.440","Text":"we have that x is smaller than 0,"},{"Start":"02:41.440 ","End":"02:44.325","Text":"up to minus Delta."},{"Start":"02:44.325 ","End":"02:47.780","Text":"Part a first. In part a,"},{"Start":"02:47.780 ","End":"02:50.210","Text":"we\u0027ll be using the upper part."},{"Start":"02:50.210 ","End":"02:54.990","Text":"Notice in the picture that this branch is part a"},{"Start":"02:54.990 ","End":"02:58.665","Text":"where x is bigger than 0,"},{"Start":"02:58.665 ","End":"03:02.345","Text":"and so we have that f of x,"},{"Start":"03:02.345 ","End":"03:05.435","Text":"which is y, is equal to 1."},{"Start":"03:05.435 ","End":"03:07.700","Text":"Here in part b,"},{"Start":"03:07.700 ","End":"03:13.255","Text":"we have that f of x is equal to minus 1."},{"Start":"03:13.255 ","End":"03:15.200","Text":"In part a,"},{"Start":"03:15.200 ","End":"03:18.890","Text":"we get from this Epsilon inequality"},{"Start":"03:18.890 ","End":"03:24.430","Text":"that the absolute value of f of x is just 1,"},{"Start":"03:24.430 ","End":"03:28.625","Text":"and L is also 1,"},{"Start":"03:28.625 ","End":"03:33.070","Text":"absolute value of 1 minus 1 less than Epsilon."},{"Start":"03:33.070 ","End":"03:38.260","Text":"This gives us that 0 is less than Epsilon."},{"Start":"03:38.260 ","End":"03:41.835","Text":"This is true always,"},{"Start":"03:41.835 ","End":"03:45.430","Text":"you don\u0027t have to even put any conditions on Delta."},{"Start":"03:45.430 ","End":"03:49.570","Text":"Actually you can take any positive Delta you like."},{"Start":"03:49.570 ","End":"03:52.495","Text":"I could tell you take Delta equals 1, but,"},{"Start":"03:52.495 ","End":"03:54.640","Text":"you know, take anything you want,"},{"Start":"03:54.640 ","End":"03:56.545","Text":"and this will be true."},{"Start":"03:56.545 ","End":"04:00.955","Text":"Similarly in part B, in part B,"},{"Start":"04:00.955 ","End":"04:04.060","Text":"the f of x minus L less than Epsilon,"},{"Start":"04:04.060 ","End":"04:08.045","Text":"F of x in part B is minus 1."},{"Start":"04:08.045 ","End":"04:12.390","Text":"L is also minus 1,"},{"Start":"04:12.390 ","End":"04:17.955","Text":"so it\u0027s minus minus 1 less than Epsilon."},{"Start":"04:17.955 ","End":"04:25.800","Text":"That just gives us also that 0 is less than Epsilon,"},{"Start":"04:25.800 ","End":"04:28.250","Text":"which is also always true,"},{"Start":"04:28.250 ","End":"04:31.865","Text":"regardless of what we take Delta to be."},{"Start":"04:31.865 ","End":"04:35.150","Text":"It\u0027s a trivial case here,"},{"Start":"04:35.150 ","End":"04:42.120","Text":"just write Delta equals anything positive."},{"Start":"04:42.120 ","End":"04:46.950","Text":"Then this will work. That\u0027s all."}],"ID":8319},{"Watched":false,"Name":"Exercise 10","Duration":"3m 6s","ChapterTopicVideoID":8166,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"In this exercise, we have to use the definition"},{"Start":"00:02.790 ","End":"00:06.420","Text":"of the limit to prove this limit."},{"Start":"00:06.420 ","End":"00:09.870","Text":"Its a limit which is equal to minus infinity,"},{"Start":"00:09.870 ","End":"00:12.450","Text":"and that has a special definition."},{"Start":"00:12.450 ","End":"00:19.380","Text":"The definition when x goes to a of f of x is minus infinity,"},{"Start":"00:19.380 ","End":"00:25.844","Text":"and it\u0027s as follows: if N is bigger than 0,"},{"Start":"00:25.844 ","End":"00:28.710","Text":"then there is a corresponding Delta"},{"Start":"00:28.710 ","End":"00:30.780","Text":"such that f of x is less than"},{"Start":"00:30.780 ","End":"00:36.265","Text":"N whenever x is close to a within Delta,"},{"Start":"00:36.265 ","End":"00:38.520","Text":"but not equal to a."},{"Start":"00:38.520 ","End":"00:41.660","Text":"In our case as I say, this a is 2,"},{"Start":"00:41.660 ","End":"00:44.525","Text":"f of x is this whole expression."},{"Start":"00:44.525 ","End":"00:47.435","Text":"Let\u0027s see, we usually start from here,"},{"Start":"00:47.435 ","End":"00:49.130","Text":"even though the logical implication"},{"Start":"00:49.130 ","End":"00:51.740","Text":"of the whenever means that this implies this,"},{"Start":"00:51.740 ","End":"00:54.750","Text":"still, we start from here,"},{"Start":"00:54.750 ","End":"00:56.945","Text":"and we work our way backwards."},{"Start":"00:56.945 ","End":"01:03.215","Text":"F of x less than N means the minus 5 over x minus 2"},{"Start":"01:03.215 ","End":"01:08.570","Text":"squared is less than N. Now we\u0027re going to do"},{"Start":"01:08.570 ","End":"01:11.180","Text":"a bit of algebra here and inequalities."},{"Start":"01:11.180 ","End":"01:17.525","Text":"If I multiply both sides by x minus 2 squared,"},{"Start":"01:17.525 ","End":"01:22.130","Text":"I\u0027ve got minus 5 is less than N,"},{"Start":"01:22.130 ","End":"01:25.315","Text":"x minus 2 squared."},{"Start":"01:25.315 ","End":"01:28.725","Text":"Now I\u0027m going to divide by N,"},{"Start":"01:28.725 ","End":"01:31.180","Text":"but N is negative,"},{"Start":"01:31.180 ","End":"01:34.655","Text":"so I\u0027m going to switch the direction of the inequality,"},{"Start":"01:34.655 ","End":"01:41.945","Text":"minus 5/N is bigger than x minus 2 squared."},{"Start":"01:41.945 ","End":"01:45.890","Text":"Now I take the square root of both sides,"},{"Start":"01:45.890 ","End":"01:49.190","Text":"and I\u0027m also going to flip sides."},{"Start":"01:49.190 ","End":"01:50.630","Text":"I\u0027m going to write this over here,"},{"Start":"01:50.630 ","End":"01:53.060","Text":"this over here, I\u0027ll take the square root."},{"Start":"01:53.060 ","End":"01:58.400","Text":"We have absolute value of x minus 2."},{"Start":"01:58.400 ","End":"01:59.960","Text":"Absolute value is because remember"},{"Start":"01:59.960 ","End":"02:00.950","Text":"that the square root of a"},{"Start":"02:00.950 ","End":"02:04.235","Text":"squared is absolute value of a in general."},{"Start":"02:04.235 ","End":"02:09.715","Text":"This is less than the square root of minus 5/N."},{"Start":"02:09.715 ","End":"02:13.805","Text":"Remember that N is negative and the square root is okay."},{"Start":"02:13.805 ","End":"02:15.830","Text":"Now if I look at this,"},{"Start":"02:15.830 ","End":"02:17.060","Text":"and I look at this,"},{"Start":"02:17.060 ","End":"02:20.195","Text":"it seems pretty straightforward that if I take"},{"Start":"02:20.195 ","End":"02:26.785","Text":"my Delta equal to square root of minus 5/N,"},{"Start":"02:26.785 ","End":"02:28.800","Text":"then that\u0027ll do the trick."},{"Start":"02:28.800 ","End":"02:31.865","Text":"Because if x minus a,"},{"Start":"02:31.865 ","End":"02:34.085","Text":"which is x minus 2,"},{"Start":"02:34.085 ","End":"02:37.955","Text":"is between Delta and 0, well,"},{"Start":"02:37.955 ","End":"02:40.550","Text":"we don\u0027t actually need the bigger than 0 part,"},{"Start":"02:40.550 ","End":"02:42.215","Text":"but if this is true,"},{"Start":"02:42.215 ","End":"02:44.915","Text":"then since Delta is this,"},{"Start":"02:44.915 ","End":"02:47.040","Text":"then this is true,"},{"Start":"02:47.040 ","End":"02:52.160","Text":"and everything else just works its way backwards,"},{"Start":"02:52.160 ","End":"02:57.620","Text":"so we get that f of x is less than N."},{"Start":"02:57.620 ","End":"02:59.885","Text":"I just like to highlight the important part,"},{"Start":"02:59.885 ","End":"03:01.310","Text":"that given N,"},{"Start":"03:01.310 ","End":"03:03.740","Text":"what is the Delta that we take?"},{"Start":"03:03.740 ","End":"03:06.930","Text":"This is it. Now we\u0027re done."}],"ID":8320},{"Watched":false,"Name":"Exercise 11","Duration":"3m 8s","ChapterTopicVideoID":8167,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.905","Text":"Here we have another limit to prove using the definition."},{"Start":"00:04.905 ","End":"00:08.625","Text":"There are 2 important things to note here,"},{"Start":"00:08.625 ","End":"00:12.060","Text":"that the limit is infinity and that"},{"Start":"00:12.060 ","End":"00:14.160","Text":"it requires a special definition."},{"Start":"00:14.160 ","End":"00:18.000","Text":"We also have to make an adjustment for the 1-sided limit."},{"Start":"00:18.000 ","End":"00:23.950","Text":"It\u0027s the limit from below or from the left."},{"Start":"00:26.210 ","End":"00:31.200","Text":"In general, we would have a function f of x"},{"Start":"00:31.200 ","End":"00:36.435","Text":"and the limit would be x goes to a,"},{"Start":"00:36.435 ","End":"00:37.980","Text":"I guess I should also here,"},{"Start":"00:37.980 ","End":"00:42.570","Text":"have said from the left."},{"Start":"00:42.730 ","End":"00:45.365","Text":"Because of the infinity,"},{"Start":"00:45.365 ","End":"00:47.240","Text":"we have the M,"},{"Start":"00:47.240 ","End":"00:50.510","Text":"and we say that for m bigger than 0,"},{"Start":"00:50.510 ","End":"00:55.520","Text":"no matter how big there is a Delta such that"},{"Start":"00:55.520 ","End":"00:58.150","Text":"f of x is bigger than m whenever,"},{"Start":"00:58.150 ","End":"01:00.020","Text":"and then this is the special condition"},{"Start":"01:00.020 ","End":"01:01.985","Text":"because it\u0027s a limit from the left."},{"Start":"01:01.985 ","End":"01:04.955","Text":"It\u0027s not our usual absolute value and so on."},{"Start":"01:04.955 ","End":"01:08.765","Text":"It\u0027s this between 0 and minus Delta."},{"Start":"01:08.765 ","End":"01:13.580","Text":"In our case, this would be x minus 3"},{"Start":"01:13.580 ","End":"01:16.875","Text":"negative but bigger than minus Delta."},{"Start":"01:16.875 ","End":"01:20.045","Text":"Let\u0027s see what this inequality is."},{"Start":"01:20.045 ","End":"01:25.805","Text":"This says that minus 2 over x"},{"Start":"01:25.805 ","End":"01:32.420","Text":"minus 3 is bigger than M. Now in general,"},{"Start":"01:32.420 ","End":"01:35.915","Text":"we\u0027re going to be working our way back."},{"Start":"01:35.915 ","End":"01:38.735","Text":"Whenever it means that this implies this."},{"Start":"01:38.735 ","End":"01:40.640","Text":"But we start here and work our way"},{"Start":"01:40.640 ","End":"01:44.925","Text":"backwards to get to Delta."},{"Start":"01:44.925 ","End":"01:48.250","Text":"Now if this is true,"},{"Start":"01:48.250 ","End":"01:51.800","Text":"then x minus 3 is negative."},{"Start":"01:51.800 ","End":"01:54.110","Text":"Note this bit."},{"Start":"01:54.110 ","End":"01:58.910","Text":"If we multiply both sides by something negative,"},{"Start":"01:58.910 ","End":"02:00.785","Text":"we invert the inequality."},{"Start":"02:00.785 ","End":"02:07.440","Text":"We\u0027ve got minus 2 is less than M times x minus 3."},{"Start":"02:07.440 ","End":"02:09.050","Text":"But M is positive,"},{"Start":"02:09.050 ","End":"02:11.905","Text":"so we can divide by it like so."},{"Start":"02:11.905 ","End":"02:14.630","Text":"Now if I look at this and I look at this,"},{"Start":"02:14.630 ","End":"02:22.405","Text":"it seems obvious that we should let Delta be equal to 2/M."},{"Start":"02:22.405 ","End":"02:25.805","Text":"Certainly Delta is positive because M is positive,"},{"Start":"02:25.805 ","End":"02:30.920","Text":"and then we can see that from here that this just says"},{"Start":"02:30.920 ","End":"02:36.920","Text":"that minus Delta is less than x minus 3."},{"Start":"02:36.920 ","End":"02:41.510","Text":"If I started off from this and said x minus 3"},{"Start":"02:41.510 ","End":"02:43.795","Text":"between minus Delta and 0,"},{"Start":"02:43.795 ","End":"02:48.530","Text":"and since Delta is 2 over m, this implies this,"},{"Start":"02:48.530 ","End":"02:53.715","Text":"this, this, which is just what we have here."},{"Start":"02:53.715 ","End":"02:56.595","Text":"We start from here and end here,"},{"Start":"02:56.595 ","End":"03:00.125","Text":"and as usual, I like to highlight the important bit,"},{"Start":"03:00.125 ","End":"03:02.330","Text":"which is what Delta is in terms of M"},{"Start":"03:02.330 ","End":"03:04.565","Text":"or Epsilon in the other cases,"},{"Start":"03:04.565 ","End":"03:09.330","Text":"and that\u0027s it. We\u0027ve proven it."}],"ID":8321},{"Watched":false,"Name":"Exercise 12","Duration":"3m 33s","ChapterTopicVideoID":8168,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.710","Text":"Here we have another limit to prove"},{"Start":"00:01.710 ","End":"00:04.065","Text":"using the definition of the limit."},{"Start":"00:04.065 ","End":"00:08.160","Text":"In this case, we have the limit as x goes to"},{"Start":"00:08.160 ","End":"00:11.550","Text":"0 from above or from the right of natural log"},{"Start":"00:11.550 ","End":"00:13.394","Text":"of x is minus infinity."},{"Start":"00:13.394 ","End":"00:16.305","Text":"It\u0027s actually a famous limit."},{"Start":"00:16.305 ","End":"00:21.295","Text":"The moment I\u0027ll remind you the definition of this special limit."},{"Start":"00:21.295 ","End":"00:24.210","Text":"Just want to say that in general,"},{"Start":"00:24.210 ","End":"00:27.070","Text":"we would have some function f of x here,"},{"Start":"00:27.070 ","End":"00:28.845","Text":"and this, we would call a."},{"Start":"00:28.845 ","End":"00:31.935","Text":"Actually, I should write the plus also,"},{"Start":"00:31.935 ","End":"00:33.975","Text":"that\u0027s also a special thing."},{"Start":"00:33.975 ","End":"00:35.550","Text":"In this case,"},{"Start":"00:35.550 ","End":"00:42.150","Text":"the definition of limit is that given any"},{"Start":"00:42.150 ","End":"00:45.085","Text":"negative n as small as you like,"},{"Start":"00:45.085 ","End":"00:50.960","Text":"there is a Delta corresponding to it in such a way that f of"},{"Start":"00:50.960 ","End":"00:56.510","Text":"x is less than n. Whenever this inequality holds for x,"},{"Start":"00:56.510 ","End":"00:58.940","Text":"it\u0027s not a usual 1,"},{"Start":"00:58.940 ","End":"01:01.790","Text":"it\u0027s because of the limit from the right that"},{"Start":"01:01.790 ","End":"01:05.485","Text":"x minus a has to be bigger than 0 but less than Delta."},{"Start":"01:05.485 ","End":"01:09.080","Text":"But sometimes, it\u0027s more convenient to write this as"},{"Start":"01:09.080 ","End":"01:13.790","Text":"x less than a plus Delta and bigger than a."},{"Start":"01:13.790 ","End":"01:16.225","Text":"We\u0027ll see which of these forms is better for us."},{"Start":"01:16.225 ","End":"01:18.990","Text":"Now, in general, whenever something,"},{"Start":"01:18.990 ","End":"01:21.195","Text":"and this implies this,"},{"Start":"01:21.195 ","End":"01:23.740","Text":"but for practicality, we start from here"},{"Start":"01:23.740 ","End":"01:26.865","Text":"and work our way back to finding Delta."},{"Start":"01:26.865 ","End":"01:29.950","Text":"Let\u0027s start with f of x less than n which"},{"Start":"01:29.950 ","End":"01:32.229","Text":"means that the natural log"},{"Start":"01:32.229 ","End":"01:39.320","Text":"of x is less than n and n is some very negative number."},{"Start":"01:40.520 ","End":"01:43.920","Text":"Note that x is positive, at least, it\u0027s implied."},{"Start":"01:43.920 ","End":"01:47.290","Text":"When I take x going to 0 from the right,"},{"Start":"01:47.290 ","End":"01:50.890","Text":"it\u0027s implied that x is positive so we don\u0027t have"},{"Start":"01:50.890 ","End":"01:54.080","Text":"a problem with the natural logarithm."},{"Start":"01:54.080 ","End":"01:59.170","Text":"I\u0027d also like to rewrite this because we know that a is 0,"},{"Start":"01:59.170 ","End":"02:07.410","Text":"this will just come out to be 0 less than x less than Delta."},{"Start":"02:07.820 ","End":"02:10.430","Text":"Let\u0027s see if we can find Delta,"},{"Start":"02:10.430 ","End":"02:13.715","Text":"I can take the exponent of both sides."},{"Start":"02:13.715 ","End":"02:16.040","Text":"The exponent is an increasing function."},{"Start":"02:16.040 ","End":"02:21.950","Text":"We would get that x is less than e to the power of n."},{"Start":"02:21.950 ","End":"02:24.035","Text":"If I take e to the power of this,"},{"Start":"02:24.035 ","End":"02:27.260","Text":"it\u0027s x and e to the power of this is this,"},{"Start":"02:27.260 ","End":"02:29.510","Text":"n will be something,"},{"Start":"02:29.510 ","End":"02:32.915","Text":"I don\u0027t know, minus a million or something."},{"Start":"02:32.915 ","End":"02:37.670","Text":"e to the n could be very small but it will be positive."},{"Start":"02:37.670 ","End":"02:41.975","Text":"Most obvious thing to do is to let this be the Delta,"},{"Start":"02:41.975 ","End":"02:47.780","Text":"so we\u0027ll take Delta equals e to the n."},{"Start":"02:47.780 ","End":"02:48.890","Text":"Let me highlight this already,"},{"Start":"02:48.890 ","End":"02:51.270","Text":"I usually forget at the end."},{"Start":"02:51.470 ","End":"02:55.410","Text":"That basically does it because well,"},{"Start":"02:55.410 ","End":"02:57.030","Text":"we couldn\u0027t look at it backwards."},{"Start":"02:57.030 ","End":"03:02.360","Text":"If 0 is less than x less than Delta,"},{"Start":"03:02.360 ","End":"03:04.700","Text":"then x less than Delta."},{"Start":"03:04.700 ","End":"03:06.770","Text":"But Delta is e to the n means that x"},{"Start":"03:06.770 ","End":"03:10.860","Text":"is less than e to the n. This gives us this."},{"Start":"03:10.860 ","End":"03:12.675","Text":"If this is true,"},{"Start":"03:12.675 ","End":"03:17.480","Text":"then this is also true if we take the natural logarithm"},{"Start":"03:17.480 ","End":"03:19.790","Text":"of both sides and x is positive,"},{"Start":"03:19.790 ","End":"03:22.370","Text":"so that\u0027s okay, so we get to this."},{"Start":"03:22.370 ","End":"03:24.290","Text":"This is just this."},{"Start":"03:24.290 ","End":"03:27.770","Text":"We\u0027ve got from here to here just fine provided"},{"Start":"03:27.770 ","End":"03:33.100","Text":"that Delta is e to the n. We\u0027re done."}],"ID":8322},{"Watched":false,"Name":"Exercise 13","Duration":"2m 22s","ChapterTopicVideoID":8169,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we have to use the definition of"},{"Start":"00:03.510 ","End":"00:06.930","Text":"the limit to prove the following limit."},{"Start":"00:06.930 ","End":"00:11.400","Text":"Notice that it\u0027s 1 of those x goes to infinity limits,"},{"Start":"00:11.400 ","End":"00:14.820","Text":"and it has a special definition."},{"Start":"00:14.820 ","End":"00:18.150","Text":"Just want to recognize this is part of"},{"Start":"00:18.150 ","End":"00:20.790","Text":"the general form where x goes to infinity"},{"Start":"00:20.790 ","End":"00:24.720","Text":"at some function of x is L."},{"Start":"00:24.720 ","End":"00:30.870","Text":"The definition for the general case is what\u0027s written here."},{"Start":"00:30.870 ","End":"00:32.850","Text":"In other words, if we\u0027re given an Epsilon,"},{"Start":"00:32.850 ","End":"00:37.710","Text":"we have to find M such that f of x is"},{"Start":"00:37.710 ","End":"00:40.160","Text":"close to L with an Epsilon whenever"},{"Start":"00:40.160 ","End":"00:43.585","Text":"x is bigger than M. Now,"},{"Start":"00:43.585 ","End":"00:47.640","Text":"logically, this implies this,"},{"Start":"00:47.640 ","End":"00:53.370","Text":"but we start from here and work our way backwards to this."},{"Start":"00:54.260 ","End":"00:57.020","Text":"F of x minus L,"},{"Start":"00:57.020 ","End":"00:58.910","Text":"an absolute value less than Epsilon."},{"Start":"00:58.910 ","End":"01:02.550","Text":"F of x is this and L is this."},{"Start":"01:02.600 ","End":"01:05.775","Text":"This is what we have."},{"Start":"01:05.775 ","End":"01:11.750","Text":"Now we\u0027re going to simplify inside the absolute value."},{"Start":"01:11.750 ","End":"01:15.620","Text":"Common denominator, put it all over x plus 2."},{"Start":"01:15.620 ","End":"01:18.085","Text":"Now simplify the numerator."},{"Start":"01:18.085 ","End":"01:20.405","Text":"This is what we get."},{"Start":"01:20.405 ","End":"01:25.780","Text":"The plan is now to make this expression larger but simpler,"},{"Start":"01:25.780 ","End":"01:29.225","Text":"and to make it easier to extract x as bigger than something,"},{"Start":"01:29.225 ","End":"01:32.180","Text":"and that will be M. Making it larger could"},{"Start":"01:32.180 ","End":"01:35.990","Text":"be making the numerator larger or the denominator smaller."},{"Start":"01:35.990 ","End":"01:37.580","Text":"In this case, what I had in mind was"},{"Start":"01:37.580 ","End":"01:39.440","Text":"shrinking the denominator."},{"Start":"01:39.440 ","End":"01:43.790","Text":"Instead of x plus 2, I\u0027ll put x. X is positive"},{"Start":"01:43.790 ","End":"01:48.545","Text":"because it\u0027s bigger than M and M is positive."},{"Start":"01:48.545 ","End":"01:50.825","Text":"Actually I can drop the absolute value."},{"Start":"01:50.825 ","End":"01:52.970","Text":"Then 5 over x bigger than Epsilon,"},{"Start":"01:52.970 ","End":"01:54.500","Text":"if we switch sides,"},{"Start":"01:54.500 ","End":"01:59.565","Text":"we get that x is bigger than 5 over Epsilon."},{"Start":"01:59.565 ","End":"02:01.590","Text":"We\u0027re looking for x bigger than something,"},{"Start":"02:01.590 ","End":"02:03.760","Text":"and if we call this M,"},{"Start":"02:03.760 ","End":"02:05.825","Text":"then this will do the trick."},{"Start":"02:05.825 ","End":"02:08.270","Text":"We start off with x is bigger than M,"},{"Start":"02:08.270 ","End":"02:11.510","Text":"and all these steps go backwards until"},{"Start":"02:11.510 ","End":"02:15.335","Text":"we get to f of x minus L less than Epsilon."},{"Start":"02:15.335 ","End":"02:18.510","Text":"That\u0027s what we needed to prove."},{"Start":"02:18.640 ","End":"02:22.920","Text":"Anyway, we are done."}],"ID":8323},{"Watched":false,"Name":"Exercise 14","Duration":"2m 24s","ChapterTopicVideoID":8170,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this exercise, we have to use the"},{"Start":"00:03.210 ","End":"00:05.940","Text":"definition of the limit to prove"},{"Start":"00:05.940 ","End":"00:08.790","Text":"that the limit as x goes to infinity of this"},{"Start":"00:08.790 ","End":"00:12.090","Text":"expression is minus 2."},{"Start":"00:12.090 ","End":"00:16.845","Text":"There\u0027s a special definition for when x goes to infinity,"},{"Start":"00:16.845 ","End":"00:20.580","Text":"which relates to the general case where we have f of x"},{"Start":"00:20.580 ","End":"00:24.460","Text":"here and L here and the definition says,"},{"Start":"00:24.460 ","End":"00:27.359","Text":"well this, you should be familiar with this, but basically,"},{"Start":"00:27.359 ","End":"00:30.045","Text":"given Epsilon, we have to find M,"},{"Start":"00:30.045 ","End":"00:35.190","Text":"so that when this is true,"},{"Start":"00:35.190 ","End":"00:36.660","Text":"when x is bigger than M,"},{"Start":"00:36.660 ","End":"00:40.610","Text":"then f of x minus L less than Epsilon,"},{"Start":"00:40.610 ","End":"00:42.935","Text":"that whenever means that this implies this."},{"Start":"00:42.935 ","End":"00:44.840","Text":"What we do in this case is,"},{"Start":"00:44.840 ","End":"00:49.400","Text":"we work back as we start from the end and we go in"},{"Start":"00:49.400 ","End":"00:54.530","Text":"steps and finally get to x bigger than something,"},{"Start":"00:54.530 ","End":"00:57.380","Text":"and we call that something M. But logically,"},{"Start":"00:57.380 ","End":"00:59.765","Text":"it has to go from here to here."},{"Start":"00:59.765 ","End":"01:02.310","Text":"Anyway, let\u0027s start,"},{"Start":"01:02.390 ","End":"01:05.550","Text":"f of x minus L less than Epsilon,"},{"Start":"01:05.550 ","End":"01:07.760","Text":"now interpret this in our case,"},{"Start":"01:07.760 ","End":"01:11.750","Text":"just substituted f of x and L and now"},{"Start":"01:11.750 ","End":"01:14.400","Text":"we want to simplify a bit."},{"Start":"01:15.050 ","End":"01:21.190","Text":"Common denominator 2x plus 1,"},{"Start":"01:21.430 ","End":"01:23.930","Text":"collect terms in the numerator,"},{"Start":"01:23.930 ","End":"01:25.760","Text":"the 4x and the 4x cancel,"},{"Start":"01:25.760 ","End":"01:27.740","Text":"3 and 2 is 5."},{"Start":"01:27.740 ","End":"01:32.970","Text":"Now what we want to do is make it simpler."},{"Start":"01:32.970 ","End":"01:35.670","Text":"The 2x plus 1 is not so simple,"},{"Start":"01:35.670 ","End":"01:37.705","Text":"I\u0027d rather have just 2x."},{"Start":"01:37.705 ","End":"01:42.390","Text":"Now, if I replace this expression by something bigger,"},{"Start":"01:42.500 ","End":"01:45.720","Text":"then if that something bigger is less than Epsilon,"},{"Start":"01:45.720 ","End":"01:48.530","Text":"this will be less than Epsilon."},{"Start":"01:48.530 ","End":"01:50.960","Text":"To make the expression bigger,"},{"Start":"01:50.960 ","End":"01:52.640","Text":"I could make the denominator smaller."},{"Start":"01:52.640 ","End":"01:55.560","Text":"In other words, if I just put 2x here,"},{"Start":"01:55.560 ","End":"01:57.780","Text":"so the 2x plus 1,"},{"Start":"01:57.780 ","End":"02:00.450","Text":"then this implies this."},{"Start":"02:00.450 ","End":"02:07.805","Text":"x is positive here so we can drop the absolute value."},{"Start":"02:07.805 ","End":"02:10.610","Text":"Then we could extract x from the inequality,"},{"Start":"02:10.610 ","End":"02:14.075","Text":"multiply by x and divide by Epsilon."},{"Start":"02:14.075 ","End":"02:16.640","Text":"This is the inequality we have and that\u0027s good."},{"Start":"02:16.640 ","End":"02:17.840","Text":"x is bigger than something,"},{"Start":"02:17.840 ","End":"02:20.075","Text":"call that something M,"},{"Start":"02:20.075 ","End":"02:25.110","Text":"and we\u0027re done. Okay."}],"ID":8324},{"Watched":false,"Name":"Exercise 15","Duration":"2m 59s","ChapterTopicVideoID":8171,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.525","Text":"Here we have to use the definition of the limit to prove this limit,"},{"Start":"00:06.525 ","End":"00:10.680","Text":"which is of the variety x goes to infinity."},{"Start":"00:10.680 ","End":"00:15.360","Text":"In general, there is a definition for x goes to"},{"Start":"00:15.360 ","End":"00:21.720","Text":"infinity in terms of f of x and L. We just have a specific case of that."},{"Start":"00:21.720 ","End":"00:25.890","Text":"The definition is, what\u0027s written here."},{"Start":"00:25.890 ","End":"00:32.010","Text":"Given Epsilon, we have to find M such that this is true."},{"Start":"00:32.010 ","End":"00:33.300","Text":"Whenever this is true,"},{"Start":"00:33.300 ","End":"00:36.150","Text":"in other words, this implies this."},{"Start":"00:36.150 ","End":"00:40.710","Text":"But we start from here and work our way backwards to this."},{"Start":"00:40.710 ","End":"00:42.685","Text":"Let\u0027s do that."},{"Start":"00:42.685 ","End":"00:46.430","Text":"Just rewriting this in terms of what we have."},{"Start":"00:46.430 ","End":"00:49.630","Text":"F of x is this. That\u0027s here."},{"Start":"00:49.630 ","End":"00:51.630","Text":"L is 3,"},{"Start":"00:51.630 ","End":"00:53.965","Text":"and that\u0027s here less than Epsilon."},{"Start":"00:53.965 ","End":"01:00.840","Text":"Now we\u0027re going to start doing some algebra on this common denominator."},{"Start":"01:00.840 ","End":"01:03.240","Text":"Just notice minus 3 times this,"},{"Start":"01:03.240 ","End":"01:05.340","Text":"is this, this and this."},{"Start":"01:05.340 ","End":"01:08.445","Text":"I\u0027m going to tidy up the numerator."},{"Start":"01:08.445 ","End":"01:17.175","Text":"The 3x squared cancels and the minus 1 with the minus 1 gives us the minus 2."},{"Start":"01:17.175 ","End":"01:20.900","Text":"I\u0027d rather not work with the negative since it\u0027s an absolute value,"},{"Start":"01:20.900 ","End":"01:23.420","Text":"I just made this positive."},{"Start":"01:23.420 ","End":"01:28.015","Text":"Now the idea is to replace this by something bigger."},{"Start":"01:28.015 ","End":"01:32.280","Text":"That\u0027s something bigger is less than Epsilon and this will be less than Epsilon."},{"Start":"01:32.280 ","End":"01:34.760","Text":"But we want to make it also simpler."},{"Start":"01:34.760 ","End":"01:37.250","Text":"Now, making a fraction bigger,"},{"Start":"01:37.250 ","End":"01:40.880","Text":"you can increase the numerator and or decrease the denominator."},{"Start":"01:40.880 ","End":"01:42.665","Text":"That will do both."},{"Start":"01:42.665 ","End":"01:48.170","Text":"What I\u0027ll do is, I\u0027ll decrease the denominator by getting rid of this."},{"Start":"01:48.170 ","End":"01:49.820","Text":"That shrinks the denominator."},{"Start":"01:49.820 ","End":"01:55.175","Text":"I could increase the numerator since x is big."},{"Start":"01:55.175 ","End":"02:02.174","Text":"We could replace this 2 by 2x and that would be bigger."},{"Start":"02:02.174 ","End":"02:08.240","Text":"We have this and so if this is true, then this is true."},{"Start":"02:08.240 ","End":"02:14.350","Text":"In fact, all along, we\u0027re working backwards at each step implies the previous step."},{"Start":"02:14.350 ","End":"02:17.635","Text":"Now, this is easy to simplify,"},{"Start":"02:17.635 ","End":"02:22.004","Text":"2x plus 3x is 5x over x squared."},{"Start":"02:22.004 ","End":"02:25.220","Text":"Absolute value of 5 over x, that\u0027s an Epsilon,"},{"Start":"02:25.220 ","End":"02:30.500","Text":"but we can throw out the absolute value because x is positive."},{"Start":"02:30.500 ","End":"02:32.690","Text":"Whoops, is a typo here."},{"Start":"02:32.690 ","End":"02:35.485","Text":"It should be 5 over Epsilon."},{"Start":"02:35.485 ","End":"02:40.690","Text":"Let me write the 5 over Epsilon."},{"Start":"02:41.120 ","End":"02:46.620","Text":"That\u0027s going to be our natural choice for M. Yeah,"},{"Start":"02:46.620 ","End":"02:48.675","Text":"let me fix this here."},{"Start":"02:48.675 ","End":"02:59.350","Text":"Yeah, 5 over epsilon and that will do it. We\u0027re done."}],"ID":8325},{"Watched":false,"Name":"Exercise 16","Duration":"3m 32s","ChapterTopicVideoID":8172,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"In this exercise, we\u0027re given a function f of x,"},{"Start":"00:04.065 ","End":"00:10.410","Text":"which satisfies that its limit at infinity is minus 5."},{"Start":"00:10.410 ","End":"00:16.875","Text":"We have to show that there exists some M bigger than 0,"},{"Start":"00:16.875 ","End":"00:20.660","Text":"such that f of x is less than minus 4 whenever"},{"Start":"00:20.660 ","End":"00:24.380","Text":"x is bigger than M. What it says, in other words,"},{"Start":"00:24.380 ","End":"00:27.500","Text":"is that if at infinity we turn to minus 5,"},{"Start":"00:27.500 ","End":"00:30.260","Text":"then from a certain point onwards,"},{"Start":"00:30.260 ","End":"00:35.585","Text":"we\u0027re already below minus 4."},{"Start":"00:35.585 ","End":"00:42.915","Text":"Now I want to remind you what it means for the limit at infinity of this to be minus 5,"},{"Start":"00:42.915 ","End":"00:49.115","Text":"is that for each Epsilon bigger than naught is some number M,"},{"Start":"00:49.115 ","End":"00:53.420","Text":"such that f of x minus the limit,"},{"Start":"00:53.420 ","End":"00:56.045","Text":"which in this case is minus 5,"},{"Start":"00:56.045 ","End":"01:02.410","Text":"is less than Epsilon whenever x is bigger than M. Now I look"},{"Start":"01:02.410 ","End":"01:10.930","Text":"here and I see that minus 4 is a distance of 1 away from minus 5."},{"Start":"01:11.120 ","End":"01:16.005","Text":"I think about letting Epsilon be 1 here."},{"Start":"01:16.005 ","End":"01:18.670","Text":"If I do let Epsilon equal 1,"},{"Start":"01:18.670 ","End":"01:25.970","Text":"then this line becomes just 1 in place of the Epsilon."},{"Start":"01:26.910 ","End":"01:31.285","Text":"Now if you\u0027re good with the absolute value computations,"},{"Start":"01:31.285 ","End":"01:34.810","Text":"you might see straight away that this says that the distance of"},{"Start":"01:34.810 ","End":"01:38.380","Text":"f of x from minus 5 is less than 1,"},{"Start":"01:38.380 ","End":"01:45.400","Text":"which means that f of x has to be within plus or minus 1 of minus 5,"},{"Start":"01:45.400 ","End":"01:50.485","Text":"which means that it\u0027s between minus 4 and minus 6."},{"Start":"01:50.485 ","End":"01:52.825","Text":"This would be what we\u0027re looking for."},{"Start":"01:52.825 ","End":"01:56.020","Text":"But if you\u0027re not, then adept with absolute value,"},{"Start":"01:56.020 ","End":"01:59.105","Text":"then let\u0027s do it longhand."},{"Start":"01:59.105 ","End":"02:05.410","Text":"You recall from algebra that if the absolute value of A is less than B,"},{"Start":"02:05.410 ","End":"02:07.945","Text":"B of course is some positive number,"},{"Start":"02:07.945 ","End":"02:12.160","Text":"then A has to be between B and minus B."},{"Start":"02:12.160 ","End":"02:21.875","Text":"In our case, if we let f of x minus minus 5 to be A and 1 is B."},{"Start":"02:21.875 ","End":"02:24.190","Text":"Then what we have is this,"},{"Start":"02:24.190 ","End":"02:26.905","Text":"which is like the left-hand side of this."},{"Start":"02:26.905 ","End":"02:29.515","Text":"From here, we get this."},{"Start":"02:29.515 ","End":"02:40.250","Text":"Now if we just add minus 5 to all sides,"},{"Start":"02:40.250 ","End":"02:42.320","Text":"well, actually if we do that,"},{"Start":"02:42.320 ","End":"02:46.025","Text":"we get minus 4 less than f of x,"},{"Start":"02:46.025 ","End":"02:53.330","Text":"less than minus 6."},{"Start":"02:53.330 ","End":"02:57.855","Text":"The minus 6, of course was minus 1 minus 5,"},{"Start":"02:57.855 ","End":"03:02.450","Text":"and the minus 4 would be 1 minus 5."},{"Start":"03:02.450 ","End":"03:08.200","Text":"But if we\u0027re only interested in 1 side in this 1 here,"},{"Start":"03:08.200 ","End":"03:10.365","Text":"just like this 1 here,"},{"Start":"03:10.365 ","End":"03:12.150","Text":"then we get this,"},{"Start":"03:12.150 ","End":"03:15.855","Text":"which is just minus 4."},{"Start":"03:15.855 ","End":"03:18.905","Text":"Looking at it, whenever x is bigger than M,"},{"Start":"03:18.905 ","End":"03:23.245","Text":"we\u0027ve got f of x less than minus 4,"},{"Start":"03:23.245 ","End":"03:27.935","Text":"which could have gone straight away if you\u0027re good with your absolute values."},{"Start":"03:27.935 ","End":"03:33.810","Text":"That\u0027s all that is required. We\u0027re done."}],"ID":8326},{"Watched":false,"Name":"Exercise 17","Duration":"3m 44s","ChapterTopicVideoID":8173,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"In this exercise, we\u0027re given the function f of x,"},{"Start":"00:04.710 ","End":"00:10.260","Text":"which satisfies that its limit as x goes to minus infinity,"},{"Start":"00:10.260 ","End":"00:12.390","Text":"I\u0027ll just emphasize the minus."},{"Start":"00:12.390 ","End":"00:17.040","Text":"I think we had a similar exercise to this before with a plus infinity."},{"Start":"00:17.040 ","End":"00:20.580","Text":"Anyway, the limit at minus infinity of f of x is 5,"},{"Start":"00:20.580 ","End":"00:27.029","Text":"and we have to prove that there is some number M,"},{"Start":"00:27.029 ","End":"00:32.310","Text":"such that f of x is bigger than 4 whenever x"},{"Start":"00:32.310 ","End":"00:37.490","Text":"is less than M. Meaning that from a certain point downwards,"},{"Start":"00:37.490 ","End":"00:40.670","Text":"f of x is already bigger than 4."},{"Start":"00:40.670 ","End":"00:43.740","Text":"Ultimately, it goes to 5,"},{"Start":"00:43.740 ","End":"00:47.240","Text":"so from some point onwards it\u0027s bigger than 4"},{"Start":"00:47.240 ","End":"00:52.400","Text":"and actually would also be true to say less than 6 equally."},{"Start":"00:52.400 ","End":"00:54.635","Text":"But this is all that we\u0027re asked for."},{"Start":"00:54.635 ","End":"00:57.500","Text":"I\u0027ll just start with a reminder of what the definition of x"},{"Start":"00:57.500 ","End":"01:01.045","Text":"goes to minus infinity means in this case."},{"Start":"01:01.045 ","End":"01:04.830","Text":"Means that for each Epsilon bigger than 0,"},{"Start":"01:04.830 ","End":"01:07.590","Text":"there is some M. Actually,"},{"Start":"01:07.590 ","End":"01:12.340","Text":"this should be less than 0."},{"Start":"01:12.430 ","End":"01:17.495","Text":"Such that whenever x is less than this M,"},{"Start":"01:17.495 ","End":"01:24.180","Text":"then f of x is close to 5 within an error of Epsilon,"},{"Start":"01:24.180 ","End":"01:27.069","Text":"just as it stated."},{"Start":"01:28.490 ","End":"01:30.690","Text":"That\u0027s what it means here."},{"Start":"01:30.690 ","End":"01:33.570","Text":"In general, we\u0027d have the limit l here,"},{"Start":"01:33.570 ","End":"01:36.015","Text":"but in our case, l is 5."},{"Start":"01:36.015 ","End":"01:38.745","Text":"Epsilon can be anything."},{"Start":"01:38.745 ","End":"01:40.980","Text":"I\u0027m looking at 5 and I\u0027m looking at 4,"},{"Start":"01:40.980 ","End":"01:42.285","Text":"I see they\u0027re 1 apart."},{"Start":"01:42.285 ","End":"01:45.225","Text":"I\u0027m going to take Epsilon equal 1."},{"Start":"01:45.225 ","End":"01:49.860","Text":"If we let Epsilon equal 1 and just the same thing as here"},{"Start":"01:49.860 ","End":"01:54.415","Text":"with 1 written instead of the Epsilon."},{"Start":"01:54.415 ","End":"01:57.649","Text":"If you\u0027re good with these absolute value inequalities,"},{"Start":"01:57.649 ","End":"02:03.695","Text":"you can see straight away that this is the distance from f of x to 5."},{"Start":"02:03.695 ","End":"02:05.465","Text":"If it\u0027s less than 1,"},{"Start":"02:05.465 ","End":"02:11.840","Text":"then f of x has to be between 5 plus 1 and 5 minus 1."},{"Start":"02:11.840 ","End":"02:16.490","Text":"In other words, it is going to be between 6 and 4."},{"Start":"02:16.490 ","End":"02:20.355","Text":"What we\u0027re interested in is the 4 part,"},{"Start":"02:20.355 ","End":"02:22.560","Text":"the 6 is less interesting."},{"Start":"02:22.560 ","End":"02:24.350","Text":"We can\u0027t see this straight away."},{"Start":"02:24.350 ","End":"02:28.460","Text":"Let\u0027s use some more basic algebra as"},{"Start":"02:28.460 ","End":"02:34.910","Text":"a basic absolute value inequality that says if the absolute value of A is less than B,"},{"Start":"02:34.910 ","End":"02:36.755","Text":"or B is something positive,"},{"Start":"02:36.755 ","End":"02:42.120","Text":"then it\u0027s the same thing as saying that A is between plus and minus B."},{"Start":"02:42.120 ","End":"02:44.819","Text":"In our case here,"},{"Start":"02:44.819 ","End":"02:50.110","Text":"if we\u0027re just going to copy this and then look at it as a special case of this,"},{"Start":"02:50.110 ","End":"02:59.685","Text":"then we get that f of x minus 5 is between 1 and minus 1."},{"Start":"02:59.685 ","End":"03:03.950","Text":"Actually, I only care about one of them."},{"Start":"03:03.950 ","End":"03:09.890","Text":"I can just delete this less than 1 bit and just look at"},{"Start":"03:09.890 ","End":"03:16.175","Text":"this inequality and then add 5 to both sides like this."},{"Start":"03:16.175 ","End":"03:20.635","Text":"Then I just compute it minus 1 plus 5 is 4,"},{"Start":"03:20.635 ","End":"03:28.395","Text":"and so f of x is bigger than 4 as requested."},{"Start":"03:28.395 ","End":"03:32.445","Text":"Note that we could have got this straight away."},{"Start":"03:32.445 ","End":"03:35.270","Text":"It all depends how much practice you have had with this kind of"},{"Start":"03:35.270 ","End":"03:38.790","Text":"inequality, absolute value."},{"Start":"03:38.920 ","End":"03:44.670","Text":"That\u0027s it, we are done here."}],"ID":8327},{"Watched":false,"Name":"Exercise 18","Duration":"2m 28s","ChapterTopicVideoID":8174,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.385","Text":"In this exercise, we\u0027re given a positive function f of x,"},{"Start":"00:05.385 ","End":"00:08.130","Text":"the interval from a to infinity,"},{"Start":"00:08.130 ","End":"00:11.625","Text":"if you like this could be written as x bigger or equal to a,"},{"Start":"00:11.625 ","End":"00:17.070","Text":"and it has the property that the limit as x goes to infinity of f of x is 0."},{"Start":"00:17.070 ","End":"00:20.160","Text":"We have to prove using definition of the limit,"},{"Start":"00:20.160 ","End":"00:27.150","Text":"that the limit as x goes to infinity of the square root of f of x is also 0."},{"Start":"00:27.150 ","End":"00:33.840","Text":"Now the definition of this limit being 0 is that for each Epsilon there is"},{"Start":"00:33.840 ","End":"00:40.160","Text":"an M such that absolute value of f of x minus the limit,"},{"Start":"00:40.160 ","End":"00:41.465","Text":"in this case 0,"},{"Start":"00:41.465 ","End":"00:49.230","Text":"is less than Epsilon whenever x is bigger than M. Now Epsilon is any positive quantity,"},{"Start":"00:49.230 ","End":"00:50.925","Text":"doesn\u0027t have to be the letter Epsilon."},{"Start":"00:50.925 ","End":"00:53.490","Text":"Instead of Epsilon, we could take Epsilon squared,"},{"Start":"00:53.490 ","End":"00:55.890","Text":"let\u0027s call it say Epsilon_1."},{"Start":"00:55.890 ","End":"01:03.960","Text":"This also has a corresponding M such that whatever is here holds."},{"Start":"01:03.960 ","End":"01:08.510","Text":"Meaning that the absolute value of f of x,"},{"Start":"01:08.510 ","End":"01:11.900","Text":"I\u0027ve just dropped the 0 here,"},{"Start":"01:11.900 ","End":"01:16.115","Text":"is less than Epsilon squared or Epsilon_1,"},{"Start":"01:16.115 ","End":"01:22.340","Text":"whenever x is bigger than M. Now f of x is positive,"},{"Start":"01:22.340 ","End":"01:25.715","Text":"and I also should have mentioned that\u0027s why this limit makes sense."},{"Start":"01:25.715 ","End":"01:29.020","Text":"The square root of f of x is defined because f is positive,"},{"Start":"01:29.020 ","End":"01:30.780","Text":"we should have mentioned that earlier."},{"Start":"01:30.780 ","End":"01:32.900","Text":"Because f of x is positive,"},{"Start":"01:32.900 ","End":"01:35.975","Text":"we can drop the absolute value,"},{"Start":"01:35.975 ","End":"01:38.375","Text":"and if f of x is less than Epsilon squared,"},{"Start":"01:38.375 ","End":"01:41.449","Text":"then we can take the square roots of both sides which are positive,"},{"Start":"01:41.449 ","End":"01:45.155","Text":"and the square root of f of x is less than Epsilon."},{"Start":"01:45.155 ","End":"01:47.570","Text":"Now instead of the square root of f of x,"},{"Start":"01:47.570 ","End":"01:49.760","Text":"I can subtract 0 for sure,"},{"Start":"01:49.760 ","End":"01:54.630","Text":"and this is positive, so I can also put absolute value here."},{"Start":"01:56.660 ","End":"02:00.920","Text":"Note that this is true whenever x is bigger than M from here."},{"Start":"02:00.920 ","End":"02:03.650","Text":"The reason I write it this way is it now looks very"},{"Start":"02:03.650 ","End":"02:06.515","Text":"much like the definition of the limit,"},{"Start":"02:06.515 ","End":"02:08.930","Text":"but with the square root of f of x."},{"Start":"02:08.930 ","End":"02:11.630","Text":"That is the definition of this limit,"},{"Start":"02:11.630 ","End":"02:16.625","Text":"it means that for each Epsilon there is an M,"},{"Start":"02:16.625 ","End":"02:20.120","Text":"that whenever x is bigger than M, then this function,"},{"Start":"02:20.120 ","End":"02:23.120","Text":"the square root of f of x less 0,"},{"Start":"02:23.120 ","End":"02:24.860","Text":"which is the limit is less than Epsilon,"},{"Start":"02:24.860 ","End":"02:28.590","Text":"so we have proved it. That\u0027s it."}],"ID":8328},{"Watched":false,"Name":"Exercise 19","Duration":"3m 47s","ChapterTopicVideoID":8175,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we\u0027re given a limit,"},{"Start":"00:03.360 ","End":"00:04.660","Text":"and we don\u0027t have to prove it,"},{"Start":"00:04.660 ","End":"00:12.240","Text":"but I\u0027d still like to mention why the limit as x goes to infinity of this over this is 1."},{"Start":"00:12.240 ","End":"00:14.745","Text":"When you have a polynomial over a polynomial,"},{"Start":"00:14.745 ","End":"00:17.820","Text":"you just have to take the term"},{"Start":"00:17.820 ","End":"00:21.015","Text":"with the highest power on the numerator and the denominator."},{"Start":"00:21.015 ","End":"00:22.320","Text":"It\u0027s the x squared,"},{"Start":"00:22.320 ","End":"00:24.015","Text":"and it\u0027s the x squared,"},{"Start":"00:24.015 ","End":"00:27.485","Text":"and you take their coefficients,"},{"Start":"00:27.485 ","End":"00:30.215","Text":"if they\u0027re equal power we just divide"},{"Start":"00:30.215 ","End":"00:34.730","Text":"this coefficient by this coefficient which is 1 over 1, which is 1."},{"Start":"00:34.730 ","End":"00:37.270","Text":"Anyway, we didn\u0027t need to prove this."},{"Start":"00:37.270 ","End":"00:40.770","Text":"I\u0027ll call this function f of x,"},{"Start":"00:40.770 ","End":"00:42.575","Text":"1 is the limit."},{"Start":"00:42.575 ","End":"00:48.500","Text":"What we\u0027re asked to do is to find a value for M,"},{"Start":"00:48.500 ","End":"00:50.705","Text":"to find some number,"},{"Start":"00:50.705 ","End":"00:55.730","Text":"such that absolute value of f of x minus L,"},{"Start":"00:55.730 ","End":"00:57.965","Text":"L being the 1 here,"},{"Start":"00:57.965 ","End":"01:01.430","Text":"is less than 0.1,"},{"Start":"01:01.430 ","End":"01:08.040","Text":"which is going to be our Epsilon whenever x is bigger than M. In other words,"},{"Start":"01:08.040 ","End":"01:12.245","Text":"this is just a general Epsilon M thing."},{"Start":"01:12.245 ","End":"01:14.360","Text":"We\u0027re given a specific Epsilon,"},{"Start":"01:14.360 ","End":"01:19.530","Text":"and we have to find a specific M that corresponds to this Epsilon."},{"Start":"01:20.210 ","End":"01:23.160","Text":"Here I just wrote this out."},{"Start":"01:23.160 ","End":"01:24.720","Text":"F of x is this,"},{"Start":"01:24.720 ","End":"01:26.070","Text":"L is 1,"},{"Start":"01:26.070 ","End":"01:30.135","Text":"and 0.1 stays as is."},{"Start":"01:30.135 ","End":"01:33.265","Text":"I\u0027m just going to do a bit of algebra now."},{"Start":"01:33.265 ","End":"01:36.565","Text":"Common denominator, which is this,"},{"Start":"01:36.565 ","End":"01:38.130","Text":"so the minus 1,"},{"Start":"01:38.130 ","End":"01:42.895","Text":"is just minus all the terms in the denominator,"},{"Start":"01:42.895 ","End":"01:47.375","Text":"and then I want to simplify the numerator."},{"Start":"01:47.375 ","End":"01:48.950","Text":"The x squareds cancel,"},{"Start":"01:48.950 ","End":"01:51.170","Text":"2x minus 3x is minus x."},{"Start":"01:51.170 ","End":"01:54.745","Text":"We have minus x minus 2,"},{"Start":"01:54.745 ","End":"01:58.340","Text":"and we can throw out the minuses because it\u0027s"},{"Start":"01:58.340 ","End":"02:02.370","Text":"absolute value and let\u0027s make the numerator positive."},{"Start":"02:03.980 ","End":"02:10.940","Text":"Now, our usual trick is to replace this by something larger and if I can prove it,"},{"Start":"02:10.940 ","End":"02:15.035","Text":"something larger is less than 0.1 and this will be less than."},{"Start":"02:15.035 ","End":"02:18.430","Text":"There\u0027s two ways to make a fraction larger,"},{"Start":"02:18.430 ","End":"02:22.650","Text":"is to increase the numerator or decrease the denominator."},{"Start":"02:22.650 ","End":"02:25.069","Text":"I\u0027m going to do both."},{"Start":"02:25.070 ","End":"02:27.330","Text":"X is bigger than 1,"},{"Start":"02:27.330 ","End":"02:28.650","Text":"or we can assume it is, because I mean,"},{"Start":"02:28.650 ","End":"02:30.900","Text":"x is going to infinity."},{"Start":"02:30.900 ","End":"02:33.165","Text":"If x is bigger than 1,"},{"Start":"02:33.165 ","End":"02:36.420","Text":"then 2x is bigger than 2."},{"Start":"02:36.420 ","End":"02:38.775","Text":"I\u0027m replacing this 2 by 2x,"},{"Start":"02:38.775 ","End":"02:40.935","Text":"I\u0027ve increased the numerator."},{"Start":"02:40.935 ","End":"02:47.160","Text":"The denominator I can decrease by just throwing these last 2 terms out,"},{"Start":"02:47.160 ","End":"02:52.215","Text":"so I have x plus 2x over x squared less than 0.1."},{"Start":"02:52.215 ","End":"02:55.305","Text":"Now, I simplify this a bit,"},{"Start":"02:55.305 ","End":"02:56.985","Text":"throw out the absolute value,"},{"Start":"02:56.985 ","End":"02:59.070","Text":"because x is positive,"},{"Start":"02:59.070 ","End":"03:03.555","Text":"and then x plus 2x is 3x over x squared is 3 over x."},{"Start":"03:03.555 ","End":"03:06.120","Text":"3 over x is less than 0.1."},{"Start":"03:06.120 ","End":"03:08.005","Text":"Since x is positive,"},{"Start":"03:08.005 ","End":"03:10.610","Text":"I can bring this to the other side and divide by"},{"Start":"03:10.610 ","End":"03:16.715","Text":"0.1 and this gives us the inequality x is bigger than 30."},{"Start":"03:16.715 ","End":"03:22.085","Text":"This 30 is the M that we were looking for above."},{"Start":"03:22.085 ","End":"03:25.735","Text":"That\u0027s our answer, 30."},{"Start":"03:25.735 ","End":"03:27.950","Text":"Just notice, not a unique answer."},{"Start":"03:27.950 ","End":"03:30.380","Text":"Anything larger than 30 would also work."},{"Start":"03:30.380 ","End":"03:33.620","Text":"I could put here 50 and that would also be good."},{"Start":"03:33.620 ","End":"03:35.240","Text":"If x is bigger than 50,"},{"Start":"03:35.240 ","End":"03:38.900","Text":"then it will be bigger than 30 and then this will be true and then this will be true,"},{"Start":"03:38.900 ","End":"03:40.550","Text":"and so on and so on."},{"Start":"03:40.550 ","End":"03:48.550","Text":"I was just mentioning, but 30 is a good answer. Done."}],"ID":8329},{"Watched":false,"Name":"Exercise 20","Duration":"2m 43s","ChapterTopicVideoID":8176,"CourseChapterTopicPlaylistID":4508,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"In this exercise, we have to use the definition of continuity,"},{"Start":"00:04.890 ","End":"00:06.810","Text":"the 1 with Epsilon Delta,"},{"Start":"00:06.810 ","End":"00:12.705","Text":"to prove that this function f of x is 2x minus 3 is continuous at x equals 4."},{"Start":"00:12.705 ","End":"00:14.490","Text":"Now according to the definition,"},{"Start":"00:14.490 ","End":"00:20.835","Text":"what we have to show is that for each Epsilon,"},{"Start":"00:20.835 ","End":"00:22.485","Text":"just writing this briefly,"},{"Start":"00:22.485 ","End":"00:32.565","Text":"for each this, there exists a Delta bigger than 0 such that the absolute value"},{"Start":"00:32.565 ","End":"00:42.210","Text":"of f of x minus f of 4 is less"},{"Start":"00:42.210 ","End":"00:52.130","Text":"than Epsilon whenever absolute value of x minus 4 is less than Delta."},{"Start":"00:52.130 ","End":"00:55.160","Text":"4 is like the a in the definition here,"},{"Start":"00:55.160 ","End":"00:58.620","Text":"we have f of x and then we have x equals a."},{"Start":"00:59.240 ","End":"01:04.520","Text":"Even though this is going to imply this,"},{"Start":"01:04.520 ","End":"01:06.920","Text":"we start from here and try to reach here."},{"Start":"01:06.920 ","End":"01:10.370","Text":"Let\u0027s see, what is f of x minus f of 4?"},{"Start":"01:10.370 ","End":"01:16.875","Text":"Well, this is f of x is 2 x minus 3."},{"Start":"01:16.875 ","End":"01:20.700","Text":"F of 4 is let\u0027s see,"},{"Start":"01:20.700 ","End":"01:30.880","Text":"that\u0027s twice 4 minus 3 is 8 minus 3 is 5."},{"Start":"01:32.600 ","End":"01:37.180","Text":"It\u0027s got to be less than Epsilon."},{"Start":"01:37.940 ","End":"01:40.545","Text":"Minus 3 minus 5 is minus 8."},{"Start":"01:40.545 ","End":"01:45.930","Text":"It\u0027s absolute value of 2x minus 8 less than Epsilon."},{"Start":"01:45.930 ","End":"01:49.100","Text":"I can take 2 outside the brackets."},{"Start":"01:49.100 ","End":"01:51.000","Text":"The absolute value,"},{"Start":"01:51.000 ","End":"01:52.410","Text":"because 2 is positive,"},{"Start":"01:52.410 ","End":"01:56.865","Text":"twice x minus 4 less than Epsilon,"},{"Start":"01:56.865 ","End":"02:02.710","Text":"x minus 4 less than Epsilon over 2."},{"Start":"02:02.780 ","End":"02:08.235","Text":"If we let Epsilon over 2 equals Delta,"},{"Start":"02:08.235 ","End":"02:10.360","Text":"I really should have said it the other way round,"},{"Start":"02:10.360 ","End":"02:13.585","Text":"let Delta equal Epsilon over 2,"},{"Start":"02:13.585 ","End":"02:16.580","Text":"then we get what we needed."},{"Start":"02:16.580 ","End":"02:20.345","Text":"If x minus 4 is less than Delta was like going backwards."},{"Start":"02:20.345 ","End":"02:22.010","Text":"If this is less than Delta,"},{"Start":"02:22.010 ","End":"02:24.490","Text":"which is this, then this is true."},{"Start":"02:24.490 ","End":"02:26.735","Text":"If this is true, then this is true,"},{"Start":"02:26.735 ","End":"02:28.280","Text":"and then this is true,"},{"Start":"02:28.280 ","End":"02:29.840","Text":"and then this is true."},{"Start":"02:29.840 ","End":"02:35.660","Text":"We\u0027ve proved this direction and yeah,"},{"Start":"02:35.660 ","End":"02:38.960","Text":"Delta equals Epsilon over 2 does the trick."},{"Start":"02:38.960 ","End":"02:43.170","Text":"That\u0027s all we needed to complete the proof."}],"ID":8330}],"Thumbnail":null,"ID":4508},{"Name":"Technique 1 Substitution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Substitution","Duration":"8m 43s","ChapterTopicVideoID":2,"CourseChapterTopicPlaylistID":162,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Several techniques for computing the limit of a function."},{"Start":"00:03.570 ","End":"00:06.645","Text":"In this clip we\u0027ll be d1emonstrating the first 1,"},{"Start":"00:06.645 ","End":"00:09.565","Text":"which is known as substitution."},{"Start":"00:09.565 ","End":"00:16.530","Text":"What this basically says is that if the function is defined at the given point,"},{"Start":"00:16.530 ","End":"00:19.890","Text":"which was not the case that we had before in"},{"Start":"00:19.890 ","End":"00:24.345","Text":"the example with x squared minus 1 over x minus 1."},{"Start":"00:24.345 ","End":"00:27.090","Text":"It leaves the function is defined at a given point that in"},{"Start":"00:27.090 ","End":"00:30.105","Text":"order to compute the limit of the function at this point,"},{"Start":"00:30.105 ","End":"00:33.945","Text":"we simply substitute the point in the function."},{"Start":"00:33.945 ","End":"00:36.810","Text":"I\u0027m going to show you a few examples."},{"Start":"00:36.810 ","End":"00:41.250","Text":"Then I\u0027m going to tell you that this is not quite true, but don\u0027t worry."},{"Start":"00:41.410 ","End":"00:45.740","Text":"Let\u0027s take here for examples that should do for a start,"},{"Start":"00:45.740 ","End":"00:51.755","Text":"the limit as x tends to 1 of x squared minus 4x plus 1."},{"Start":"00:51.755 ","End":"00:58.010","Text":"Now note that this function is defined for all values of x."},{"Start":"00:58.010 ","End":"01:02.780","Text":"In particular, it\u0027s defined for x is 1."},{"Start":"01:02.780 ","End":"01:05.780","Text":"In fact, in all these examples that I\u0027ve given here,"},{"Start":"01:05.780 ","End":"01:10.000","Text":"these functions are always defined at the limit value."},{"Start":"01:10.000 ","End":"01:11.795","Text":"In this case, in each,"},{"Start":"01:11.795 ","End":"01:14.240","Text":"we can use substitution and all of them."},{"Start":"01:14.240 ","End":"01:16.280","Text":"What do we do for this 1,"},{"Start":"01:16.280 ","End":"01:19.160","Text":"for example, x goes to 1."},{"Start":"01:19.160 ","End":"01:22.340","Text":"I\u0027m going to substitute instead of x,"},{"Start":"01:22.340 ","End":"01:24.840","Text":"I\u0027m going to put 1."},{"Start":"01:24.890 ","End":"01:27.620","Text":"Here I\u0027ve done that substitution."},{"Start":"01:27.620 ","End":"01:29.510","Text":"I\u0027ve put 1 instead of x,"},{"Start":"01:29.510 ","End":"01:34.475","Text":"so we get 1 squared minus 4 times 1 plus 1."},{"Start":"01:34.475 ","End":"01:43.655","Text":"That\u0027s equal to minus 2 because we have 1 minus 4 plus 1, that\u0027s minus 2."},{"Start":"01:43.655 ","End":"01:46.645","Text":"That\u0027s the answer to the first one."},{"Start":"01:46.645 ","End":"01:48.700","Text":"Now in the second one,"},{"Start":"01:48.700 ","End":"01:51.280","Text":"all I have to do is substitute x equals 4,"},{"Start":"01:51.280 ","End":"01:55.385","Text":"and we\u0027ll do that in blue in this exercise,"},{"Start":"01:55.385 ","End":"01:59.250","Text":"and we\u0027ll get this."},{"Start":"01:59.250 ","End":"02:01.445","Text":"If we compute this,"},{"Start":"02:01.445 ","End":"02:03.275","Text":"and we can do this in our heads,"},{"Start":"02:03.275 ","End":"02:08.340","Text":"a 104 minus 4 is a 100 and the square root of a 100 is 10."},{"Start":"02:09.710 ","End":"02:12.240","Text":"Yet another example."},{"Start":"02:12.240 ","End":"02:16.795","Text":"First thing is to substitute x equals 10."},{"Start":"02:16.795 ","End":"02:20.990","Text":"Notice that we substituting 10 here,"},{"Start":"02:20.990 ","End":"02:25.940","Text":"but we could have substituted anything that is anything except minus 18."},{"Start":"02:25.940 ","End":"02:29.960","Text":"This is one of those functions which is not defined everywhere,"},{"Start":"02:29.960 ","End":"02:32.265","Text":"but it is our point which is 10."},{"Start":"02:32.265 ","End":"02:40.205","Text":"We get this that x we put 10 in blue even."},{"Start":"02:40.205 ","End":"02:42.080","Text":"We compute this."},{"Start":"02:42.080 ","End":"02:44.840","Text":"10 plus 4 is 14."},{"Start":"02:44.840 ","End":"02:46.490","Text":"10 plus 18 is 28,"},{"Start":"02:46.490 ","End":"02:50.920","Text":"14 over 28 is equal to 1.5."},{"Start":"02:50.920 ","End":"02:52.845","Text":"That\u0027s the answer to this 1."},{"Start":"02:52.845 ","End":"02:54.860","Text":"The last one\u0027s a bit of a funny one."},{"Start":"02:54.860 ","End":"02:58.310","Text":"People sometimes confused by the constant function."},{"Start":"02:58.310 ","End":"03:00.290","Text":"This is the constant function 40,"},{"Start":"03:00.290 ","End":"03:03.220","Text":"which means for any x is 40."},{"Start":"03:03.220 ","End":"03:05.180","Text":"It doesn\u0027t care about X,"},{"Start":"03:05.180 ","End":"03:07.550","Text":"whatever x is, it\u0027s 40."},{"Start":"03:07.550 ","End":"03:12.425","Text":"The limit as x goes to a 140 is still 40."},{"Start":"03:12.425 ","End":"03:14.880","Text":"Those who don\u0027t quite get it,"},{"Start":"03:14.880 ","End":"03:16.310","Text":"sometimes I had an idea,"},{"Start":"03:16.310 ","End":"03:19.659","Text":"wants to explain 40 instead of 40,"},{"Start":"03:19.659 ","End":"03:24.855","Text":"I wrote it as 40 plus 0 x."},{"Start":"03:24.855 ","End":"03:27.739","Text":"Everyone was happy that x is in the function."},{"Start":"03:27.739 ","End":"03:30.130","Text":"Substitute x equals a 100,"},{"Start":"03:30.130 ","End":"03:32.940","Text":"40 plus 0 times a 100 is 40."},{"Start":"03:32.940 ","End":"03:36.285","Text":"If you\u0027re happier that way, It\u0027s okay."},{"Start":"03:36.285 ","End":"03:39.860","Text":"Up till now, we\u0027ve defined what to do,"},{"Start":"03:39.860 ","End":"03:41.435","Text":"and we\u0027ve given some examples."},{"Start":"03:41.435 ","End":"03:45.350","Text":"The only thing is that what I\u0027ve written here is not exactly true."},{"Start":"03:45.350 ","End":"03:49.990","Text":"There\u0027s a bit of some exceptions and let\u0027s take care of those now."},{"Start":"03:49.990 ","End":"03:53.545","Text":"What it says that this technique, substitution,"},{"Start":"03:53.545 ","End":"03:59.014","Text":"is valid as long as the function is defined by 1 single formula."},{"Start":"03:59.014 ","End":"04:01.220","Text":"Just like the functions in high school"},{"Start":"04:01.220 ","End":"04:06.040","Text":"where in the case it\u0027s defined by 2 or more formally,"},{"Start":"04:06.040 ","End":"04:12.025","Text":"we proceed differently exactly how we shall see in what is to come."},{"Start":"04:12.025 ","End":"04:19.720","Text":"Not at this point. I just wanted to mention that it doesn\u0027t always work the substitution."},{"Start":"04:20.330 ","End":"04:26.915","Text":"The other thing I\u0027d like to raise and people ask is,"},{"Start":"04:26.915 ","End":"04:30.910","Text":"if the substitution is just so basic, so simple,"},{"Start":"04:30.910 ","End":"04:37.175","Text":"why would anyone ever ask you to solve a limit with just a substitution?"},{"Start":"04:37.175 ","End":"04:41.110","Text":"The answer is, it usually won\u0027t happen unless you\u0027re very lucky,"},{"Start":"04:41.110 ","End":"04:43.015","Text":"you won\u0027t get such a thing in an exam."},{"Start":"04:43.015 ","End":"04:46.985","Text":"However, in the course of solving a question on limits,"},{"Start":"04:46.985 ","End":"04:49.790","Text":"maybe you\u0027d be doing some simplifications and that"},{"Start":"04:49.790 ","End":"04:56.270","Text":"changing and processing during the course of it you will get such thing."},{"Start":"04:56.270 ","End":"04:58.895","Text":"He may simplify something more complicated"},{"Start":"04:58.895 ","End":"05:01.805","Text":"to something which is just needs a substitution."},{"Start":"05:01.805 ","End":"05:07.600","Text":"It is definitely a useful technique to learn."},{"Start":"05:08.840 ","End":"05:13.819","Text":"In this exception paragraph,"},{"Start":"05:13.819 ","End":"05:18.290","Text":"I talked about single formula and multiple formulae."},{"Start":"05:18.290 ","End":"05:20.360","Text":"You might say, what does that mean?"},{"Start":"05:20.360 ","End":"05:24.570","Text":"Well, I\u0027ll illustrate in the case of an example,"},{"Start":"05:26.540 ","End":"05:36.890","Text":"so it could be defined in it cooled a piece wise fashion or in a split form."},{"Start":"05:36.890 ","End":"05:42.740","Text":"We say, the value of f of x depends on what we\u0027re x is."},{"Start":"05:42.740 ","End":"05:45.500","Text":"For x, which are bigger or equal to 5,"},{"Start":"05:45.500 ","End":"05:46.835","Text":"we define it 1 way,"},{"Start":"05:46.835 ","End":"05:48.665","Text":"namely x squared plus 1."},{"Start":"05:48.665 ","End":"05:50.990","Text":"But if X is less than 5,"},{"Start":"05:50.990 ","End":"05:52.534","Text":"then we use the other formula,"},{"Start":"05:52.534 ","End":"05:53.990","Text":"minus x plus 7."},{"Start":"05:53.990 ","End":"05:57.320","Text":"For example, if I want to know what f of 6 is,"},{"Start":"05:57.320 ","End":"05:59.555","Text":"6 is bigger or equal to 5."},{"Start":"05:59.555 ","End":"06:03.385","Text":"The answer is 6, 6 squared plus 1 or 37."},{"Start":"06:03.385 ","End":"06:06.110","Text":"F of 6 is 37. On the other hand,"},{"Start":"06:06.110 ","End":"06:08.795","Text":"f of 2, I put 2 here."},{"Start":"06:08.795 ","End":"06:11.765","Text":"Let\u0027s use this formula because 2 is smaller than 5."},{"Start":"06:11.765 ","End":"06:14.945","Text":"It\u0027s minus 2 plus 7, so it\u0027s 5."},{"Start":"06:14.945 ","End":"06:19.205","Text":"Now, the problem occurs when we have the limit."},{"Start":"06:19.205 ","End":"06:21.515","Text":"Something happens around 5."},{"Start":"06:21.515 ","End":"06:25.760","Text":"It\u0027s maybe a scene point,"},{"Start":"06:25.760 ","End":"06:28.775","Text":"seam line or transition point."},{"Start":"06:28.775 ","End":"06:30.530","Text":"If I was to ask,"},{"Start":"06:30.530 ","End":"06:39.650","Text":"what is the limit as x goes to 5 of f of x?"},{"Start":"06:39.650 ","End":"06:45.680","Text":"Then we can\u0027t use the technique of substitution because 5"},{"Start":"06:45.680 ","End":"06:51.520","Text":"is on a seam line between different areas with different formulae."},{"Start":"06:51.520 ","End":"06:59.590","Text":"That was the exception to assume we have 2 formulae and not 1."},{"Start":"07:01.690 ","End":"07:06.260","Text":"In any event, most functions are defined by"},{"Start":"07:06.260 ","End":"07:10.340","Text":"a simple formula that takes care of most of the cases."},{"Start":"07:10.340 ","End":"07:16.205","Text":"But yes, we will see later on split or piecewise-defined functions."},{"Start":"07:16.205 ","End":"07:18.275","Text":"We\u0027re basically done."},{"Start":"07:18.275 ","End":"07:22.160","Text":"In fact, we are done for those of you who wanted to be done,"},{"Start":"07:22.160 ","End":"07:26.340","Text":"but I have to add that there\u0027s still an exception to the exception."},{"Start":"07:27.640 ","End":"07:30.620","Text":"But this is also not entirely true."},{"Start":"07:30.620 ","End":"07:34.775","Text":"There are very, what we might call exotic kinds of a function."},{"Start":"07:34.775 ","End":"07:38.180","Text":"Now, you don\u0027t have to listen any further unless you\u0027re interested in."},{"Start":"07:38.180 ","End":"07:42.080","Text":"But we have very weird functions or maybe put it,"},{"Start":"07:42.080 ","End":"07:43.340","Text":"put it in a different color."},{"Start":"07:43.340 ","End":"07:46.055","Text":"Say, I don\u0027t know in the green,"},{"Start":"07:46.055 ","End":"07:48.695","Text":"it can define functions in any ways."},{"Start":"07:48.695 ","End":"07:54.815","Text":"You could define, let\u0027s say f of x is equal to positive 1."},{"Start":"07:54.815 ","End":"07:58.530","Text":"If x is a rational number,"},{"Start":"08:00.680 ","End":"08:06.830","Text":"and 0 if x is irrational in the mathematical sense,"},{"Start":"08:06.830 ","End":"08:09.120","Text":"meaning it\u0027s not a fraction."},{"Start":"08:12.020 ","End":"08:15.605","Text":"Even this is not really defined."},{"Start":"08:15.605 ","End":"08:18.050","Text":"It\u0027s defined only in terms of 2 formulae,"},{"Start":"08:18.050 ","End":"08:25.680","Text":"but it\u0027ll be entirely different than the techniques."},{"Start":"08:26.630 ","End":"08:29.200","Text":"It\u0027s just something else,"},{"Start":"08:29.200 ","End":"08:32.465","Text":"and they\u0027re a very bizarre things and functions that you might see."},{"Start":"08:32.465 ","End":"08:36.110","Text":"Anyway, that\u0027s just beyond the schedule at this point."},{"Start":"08:36.110 ","End":"08:38.150","Text":"You won\u0027t see them in Calculus 1."},{"Start":"08:38.150 ","End":"08:42.090","Text":"That\u0027s reassuring. We\u0027re done for now."}],"ID":2},{"Watched":false,"Name":"Exercise 1","Duration":"46s","ChapterTopicVideoID":1524,"CourseChapterTopicPlaylistID":162,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this exercise, we have to find the limit as x"},{"Start":"00:04.020 ","End":"00:08.670","Text":"tends to 4 of the function x squared plus x plus 1."},{"Start":"00:08.670 ","End":"00:10.965","Text":"This is an elementary function,"},{"Start":"00:10.965 ","End":"00:13.920","Text":"and so if it\u0027s defined at x equals 4,"},{"Start":"00:13.920 ","End":"00:19.920","Text":"which it is, all we have to do is substitute x equals 4 in the function."},{"Start":"00:19.920 ","End":"00:25.980","Text":"What we get is that the limit as x tends to"},{"Start":"00:25.980 ","End":"00:32.820","Text":"4 of x squared plus x plus 1 equals,"},{"Start":"00:32.820 ","End":"00:35.250","Text":"I\u0027m going to substitute 4 for x,"},{"Start":"00:35.250 ","End":"00:39.765","Text":"is 4 squared plus 4 plus 1;"},{"Start":"00:39.765 ","End":"00:44.520","Text":"16 plus 4 plus 1 gives us 21,"},{"Start":"00:44.520 ","End":"00:46.960","Text":"and that\u0027s the answer."}],"ID":1536},{"Watched":false,"Name":"Exercise 2","Duration":"47s","ChapterTopicVideoID":1525,"CourseChapterTopicPlaylistID":162,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this exercise, we have to find the limit as x"},{"Start":"00:03.390 ","End":"00:07.800","Text":"tends to 10 of the function x plus 1 over x plus 2."},{"Start":"00:07.800 ","End":"00:12.930","Text":"This is an elementary function which means that if we can substitute x equals 10,"},{"Start":"00:12.930 ","End":"00:15.150","Text":"that\u0027s also going to be the limit."},{"Start":"00:15.150 ","End":"00:18.135","Text":"Let\u0027s see if we can substitute x equals 10,"},{"Start":"00:18.135 ","End":"00:23.370","Text":"rewrite the exercise limit as x tends to 10"},{"Start":"00:23.370 ","End":"00:30.165","Text":"of x plus 1 over x plus 2 is equal."},{"Start":"00:30.165 ","End":"00:32.775","Text":"Instead of x I\u0027m going to put 10."},{"Start":"00:32.775 ","End":"00:39.315","Text":"I get 10 plus 1 over 10 plus 2."},{"Start":"00:39.315 ","End":"00:44.655","Text":"That\u0027s equal to 11 over 12."},{"Start":"00:44.655 ","End":"00:48.330","Text":"That\u0027s the answer to all there is to it."}],"ID":1537},{"Watched":false,"Name":"Exercise 3","Duration":"1m 3s","ChapterTopicVideoID":1526,"CourseChapterTopicPlaylistID":162,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, we have to find the limit as x tends to"},{"Start":"00:03.930 ","End":"00:09.150","Text":"1 of the function square root of x plus 3. Not quite."},{"Start":"00:09.150 ","End":"00:11.355","Text":"Notice that there\u0027s a little plus here."},{"Start":"00:11.355 ","End":"00:14.520","Text":"This means the limit as x goes to 1 from the right."},{"Start":"00:14.520 ","End":"00:16.200","Text":"But I\u0027ll ignore that for the moment,"},{"Start":"00:16.200 ","End":"00:18.385","Text":"and take care of that at the end."},{"Start":"00:18.385 ","End":"00:21.150","Text":"This function is an elementary function,"},{"Start":"00:21.150 ","End":"00:24.495","Text":"which means that if we can substitute x equals 1,"},{"Start":"00:24.495 ","End":"00:26.460","Text":"that will also be the limit."},{"Start":"00:26.460 ","End":"00:31.205","Text":"There\u0027s no reason why we can\u0027t substitute x equals 1 here, so let\u0027s see."},{"Start":"00:31.205 ","End":"00:38.360","Text":"I\u0027ll just put instead of x_1 as the square root of 1 plus 3 is 2."},{"Start":"00:38.360 ","End":"00:40.490","Text":"Okay, that\u0027s the regular limit."},{"Start":"00:40.490 ","End":"00:44.570","Text":"But in theory, you learned that if a function has a regular limit,"},{"Start":"00:44.570 ","End":"00:46.490","Text":"sometimes called a 2-sided limit,"},{"Start":"00:46.490 ","End":"00:50.690","Text":"and it also has a limit from the left and a limit from the right and all are equal."},{"Start":"00:50.690 ","End":"00:52.730","Text":"Just to make it precise,"},{"Start":"00:52.730 ","End":"00:57.770","Text":"I can also say that the limit as x tends to 1"},{"Start":"00:57.770 ","End":"01:04.350","Text":"from the right is also equal to 2, and we\u0027re done."}],"ID":1538},{"Watched":false,"Name":"Exercise 4","Duration":"37s","ChapterTopicVideoID":1527,"CourseChapterTopicPlaylistID":162,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"In this exercise, we have to find the limit as x tends to 100 of the function 20."},{"Start":"00:05.175 ","End":"00:06.930","Text":"This is a constant function."},{"Start":"00:06.930 ","End":"00:08.610","Text":"It\u0027s an elementary function,"},{"Start":"00:08.610 ","End":"00:10.635","Text":"and certainly we can substitute x equals"},{"Start":"00:10.635 ","End":"00:14.460","Text":"100 and it\u0027ll just give us 20 and that will be the limit."},{"Start":"00:14.460 ","End":"00:23.130","Text":"In other words, the limit as x tends to 100"},{"Start":"00:23.130 ","End":"00:32.475","Text":"of 20 is equal to the same expression where I substitute x is 100,"},{"Start":"00:32.475 ","End":"00:35.400","Text":"well the 20 doesn\u0027t care, it\u0027s just 20."},{"Start":"00:35.400 ","End":"00:38.530","Text":"That\u0027s the answer. We\u0027re done."}],"ID":1539}],"Thumbnail":null,"ID":162},{"Name":"Technique 2 Factoring","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Factoring","Duration":"6m 28s","ChapterTopicVideoID":8249,"CourseChapterTopicPlaylistID":163,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"In this clip, we talk about technique number 2 for computing a limit."},{"Start":"00:04.185 ","End":"00:07.200","Text":"It\u0027s called factoring and/or canceling."},{"Start":"00:07.200 ","End":"00:10.049","Text":"If the limit is of the form 0 over 0,"},{"Start":"00:10.049 ","End":"00:14.340","Text":"we try to break the numerator or the denominator or both,"},{"Start":"00:14.340 ","End":"00:16.110","Text":"and the factors, hopefully,"},{"Start":"00:16.110 ","End":"00:18.810","Text":"they have some common factor which we can cancel."},{"Start":"00:18.810 ","End":"00:20.970","Text":"I\u0027ll illustrate with a few examples."},{"Start":"00:20.970 ","End":"00:22.995","Text":"Let\u0027s take the first example,"},{"Start":"00:22.995 ","End":"00:30.840","Text":"the limit as x goes to 1 of x squared minus 1 over x minus 1."},{"Start":"00:30.840 ","End":"00:34.925","Text":"The first thing to do in the limit is to try the substitution method."},{"Start":"00:34.925 ","End":"00:36.710","Text":"If I put x equals 1,"},{"Start":"00:36.710 ","End":"00:38.880","Text":"then x minus 1 is 0,"},{"Start":"00:38.880 ","End":"00:41.415","Text":"x squared minus 1 is also 0,"},{"Start":"00:41.415 ","End":"00:44.625","Text":"so we do indeed have a 0 over 0 case."},{"Start":"00:44.625 ","End":"00:47.105","Text":"In the numerator, we have x squared minus 1,"},{"Start":"00:47.105 ","End":"00:51.620","Text":"and that\u0027s easily factored using the difference of squares formula."},{"Start":"00:51.620 ","End":"00:55.770","Text":"We end up getting the limit as x goes to 1,"},{"Start":"00:55.770 ","End":"01:00.425","Text":"this thing becomes x plus 1, x minus 1,"},{"Start":"01:00.425 ","End":"01:03.140","Text":"and in the denominator there\u0027s nothing to factorize,"},{"Start":"01:03.140 ","End":"01:05.620","Text":"so leave it as x minus 1."},{"Start":"01:05.620 ","End":"01:08.420","Text":"This point, we noticed that something can be canceled,"},{"Start":"01:08.420 ","End":"01:11.075","Text":"x minus 1 cancels with x minus 1,"},{"Start":"01:11.075 ","End":"01:17.495","Text":"and all we\u0027re left with is the limit as x goes to 1 of x plus 1,"},{"Start":"01:17.495 ","End":"01:20.420","Text":"and here, the substitution method will work."},{"Start":"01:20.420 ","End":"01:22.325","Text":"Just put x equals 1 here,"},{"Start":"01:22.325 ","End":"01:23.810","Text":"the answer is 2."},{"Start":"01:23.810 ","End":"01:28.875","Text":"Next example, limit as x goes to 0,"},{"Start":"01:28.875 ","End":"01:35.695","Text":"x squared plus 4x over x squared minus 10x."},{"Start":"01:35.695 ","End":"01:38.855","Text":"As before, I try putting x equals 0,"},{"Start":"01:38.855 ","End":"01:41.115","Text":"x squared minus 10x comes out a 0."},{"Start":"01:41.115 ","End":"01:44.270","Text":"Basically, the x being 0 makes all the terms 0,"},{"Start":"01:44.270 ","End":"01:48.085","Text":"and we end up again with the 0 over 0 form."},{"Start":"01:48.085 ","End":"01:51.350","Text":"We try the method of factoring and canceling,"},{"Start":"01:51.350 ","End":"01:53.810","Text":"and we get the limit again,"},{"Start":"01:53.810 ","End":"01:55.250","Text":"x goes to 0."},{"Start":"01:55.250 ","End":"01:58.580","Text":"The x squared plus 4x becomes x plus 4."},{"Start":"01:58.580 ","End":"02:02.930","Text":"We just take the x outside the brackets and on the denominator,"},{"Start":"02:02.930 ","End":"02:05.570","Text":"x times x minus 10."},{"Start":"02:05.570 ","End":"02:07.100","Text":"Next step is to cancel."},{"Start":"02:07.100 ","End":"02:09.020","Text":"We see that x is common,"},{"Start":"02:09.020 ","End":"02:11.405","Text":"and I should note that x is not 0."},{"Start":"02:11.405 ","End":"02:14.615","Text":"You might think I\u0027m canceling by 0 because when x tends to 0,"},{"Start":"02:14.615 ","End":"02:16.190","Text":"x does not equal 0,"},{"Start":"02:16.190 ","End":"02:17.660","Text":"so this is okay,"},{"Start":"02:17.660 ","End":"02:24.950","Text":"and we end up with the limit as x goes to 0 of x plus 4 over x minus 10."},{"Start":"02:24.950 ","End":"02:28.090","Text":"At this point, there\u0027s no problem in substitution,"},{"Start":"02:28.090 ","End":"02:29.600","Text":"so let x equals 0,"},{"Start":"02:29.600 ","End":"02:33.260","Text":"and end up with 4 over minus 10."},{"Start":"02:33.260 ","End":"02:34.759","Text":"For those who like decimals,"},{"Start":"02:34.759 ","End":"02:37.810","Text":"I could write this as minus 0.4,"},{"Start":"02:37.810 ","End":"02:39.565","Text":"and that\u0027s this example."},{"Start":"02:39.565 ","End":"02:42.670","Text":"The next exercise is a fake exercise,"},{"Start":"02:42.670 ","End":"02:44.590","Text":"and I\u0027ll explain what I mean in a minute."},{"Start":"02:44.590 ","End":"02:50.955","Text":"Limit as x goes to 1 of twice x minus 1 times x"},{"Start":"02:50.955 ","End":"02:58.485","Text":"plus 4 over 5 times x minus 1 times x plus 7."},{"Start":"02:58.485 ","End":"03:00.495","Text":"What did I mean by fake exercise?"},{"Start":"03:00.495 ","End":"03:01.735","Text":"I mean it\u0027s too easy."},{"Start":"03:01.735 ","End":"03:03.925","Text":"Look, it\u0027s already been factored for you."},{"Start":"03:03.925 ","End":"03:06.205","Text":"You can already see that it has the common factor."},{"Start":"03:06.205 ","End":"03:08.740","Text":"Here, first you check that it\u0027s a 0 over 0,"},{"Start":"03:08.740 ","End":"03:09.850","Text":"you put x equals 1,"},{"Start":"03:09.850 ","End":"03:11.995","Text":"here 0, here 0, and so on."},{"Start":"03:11.995 ","End":"03:14.200","Text":"Then you\u0027d say my work\u0027s all cut out for me,"},{"Start":"03:14.200 ","End":"03:16.090","Text":"x minus 1 is a common factor."},{"Start":"03:16.090 ","End":"03:19.755","Text":"Cancel it, then substitute x equals 1,"},{"Start":"03:19.755 ","End":"03:21.090","Text":"and at this point,"},{"Start":"03:21.090 ","End":"03:25.370","Text":"you would just do some simple arithmetic and everything falls into place."},{"Start":"03:25.370 ","End":"03:26.570","Text":"Even the 5 cancels,"},{"Start":"03:26.570 ","End":"03:28.745","Text":"2 over 8 leaves you with 1/4."},{"Start":"03:28.745 ","End":"03:30.230","Text":"This is unrealistic,"},{"Start":"03:30.230 ","End":"03:31.310","Text":"as I said, because it is too easy."},{"Start":"03:31.310 ","End":"03:34.310","Text":"What\u0027s more likely is that the professor or whoever"},{"Start":"03:34.310 ","End":"03:37.595","Text":"made up the exercise wrote something like this for himself,"},{"Start":"03:37.595 ","End":"03:41.105","Text":"but then he multiplied it out to make it more difficult on you."},{"Start":"03:41.105 ","End":"03:43.670","Text":"He would give you this exercise,"},{"Start":"03:43.670 ","End":"03:45.410","Text":"limit as x goes to 1,"},{"Start":"03:45.410 ","End":"03:55.425","Text":"2x squared plus 6x minus 8 over 5x squared plus 30x minus 35,"},{"Start":"03:55.425 ","End":"03:58.760","Text":"and you somehow have to factor it to get to here."},{"Start":"03:58.760 ","End":"04:00.020","Text":"Here in the frame,"},{"Start":"04:00.020 ","End":"04:05.405","Text":"I\u0027ve written the technique that you use to factorize a quadratic polynomial,"},{"Start":"04:05.405 ","End":"04:11.240","Text":"ax squared plus bx plus c. What you do is you first of all go to the next line,"},{"Start":"04:11.240 ","End":"04:13.370","Text":"which is to solve the quadratic equation."},{"Start":"04:13.370 ","End":"04:14.900","Text":"You set it equal to 0,"},{"Start":"04:14.900 ","End":"04:17.660","Text":"you solve the equation with the famous formula,"},{"Start":"04:17.660 ","End":"04:20.530","Text":"I\u0027m not going to repeat it, and you get 2x\u0027s."},{"Start":"04:20.530 ","End":"04:22.080","Text":"Then you get these 2x\u0027s,"},{"Start":"04:22.080 ","End":"04:23.715","Text":"which I call x_1 and x_2,"},{"Start":"04:23.715 ","End":"04:25.575","Text":"stick them in this formula,"},{"Start":"04:25.575 ","End":"04:27.320","Text":"write this thing as a,"},{"Start":"04:27.320 ","End":"04:30.140","Text":"which is the coefficient of x squared, and then in brackets,"},{"Start":"04:30.140 ","End":"04:34.245","Text":"x minus 1 of the roots times x minus the other root."},{"Start":"04:34.245 ","End":"04:37.175","Text":"Let\u0027s apply the technique from the box to here."},{"Start":"04:37.175 ","End":"04:39.800","Text":"We have twice the quadratic polynomial,"},{"Start":"04:39.800 ","End":"04:41.445","Text":"so let\u0027s write it as a fraction."},{"Start":"04:41.445 ","End":"04:48.260","Text":"This 1, we can write according to this rule as 2 times x minus the x_1 there,"},{"Start":"04:48.260 ","End":"04:50.740","Text":"x minus the x_2 there,"},{"Start":"04:50.740 ","End":"04:52.885","Text":"where the 2 is this 2,"},{"Start":"04:52.885 ","End":"04:55.425","Text":"and the same thing on the denominator,"},{"Start":"04:55.425 ","End":"05:01.160","Text":"5x minus something, x minus something,"},{"Start":"05:01.160 ","End":"05:04.295","Text":"where the 5 here and here are the same."},{"Start":"05:04.295 ","End":"05:06.470","Text":"The next thing to do, and I\u0027m not going to do,"},{"Start":"05:06.470 ","End":"05:08.750","Text":"it is to solve the quadratic equations."},{"Start":"05:08.750 ","End":"05:13.505","Text":"I solve this thing equals 0 and I\u0027ll get 2 solutions,"},{"Start":"05:13.505 ","End":"05:15.275","Text":"1 and minus 4,"},{"Start":"05:15.275 ","End":"05:17.165","Text":"but you\u0027ll do that on your own."},{"Start":"05:17.165 ","End":"05:20.360","Text":"The same here, you solve this thing equals 0."},{"Start":"05:20.360 ","End":"05:22.235","Text":"You get the 2 solutions for x,"},{"Start":"05:22.235 ","End":"05:24.305","Text":"a 1 and minus 7,"},{"Start":"05:24.305 ","End":"05:26.690","Text":"and then at the end for the factorization,"},{"Start":"05:26.690 ","End":"05:29.045","Text":"we put the x_1 x_2 here."},{"Start":"05:29.045 ","End":"05:30.755","Text":"Here\u0027s x minus 1,"},{"Start":"05:30.755 ","End":"05:33.420","Text":"x minus minus minus 4,"},{"Start":"05:33.420 ","End":"05:35.685","Text":"and here, x minus 1,"},{"Start":"05:35.685 ","End":"05:38.010","Text":"x minus minus 7."},{"Start":"05:38.010 ","End":"05:41.310","Text":"But you should replace these minus minus,"},{"Start":"05:41.310 ","End":"05:43.110","Text":"you put it as a plus, of course."},{"Start":"05:43.110 ","End":"05:47.000","Text":"All that remains to be done is to continue from this point."},{"Start":"05:47.000 ","End":"05:50.225","Text":"This is this and we just continue, it\u0027s not the answer."},{"Start":"05:50.225 ","End":"05:53.090","Text":"As you see, it\u0027s quite a lot of work to get from"},{"Start":"05:53.090 ","End":"05:56.485","Text":"this limit here all the way to the answer,"},{"Start":"05:56.485 ","End":"06:01.400","Text":"and there\u0027s extra work that I didn\u0027t do of solving 2 quadratic equations."},{"Start":"06:01.400 ","End":"06:04.220","Text":"But I have some good news that in the future,"},{"Start":"06:04.220 ","End":"06:07.925","Text":"you\u0027ll be learning a new rule that will help enormously."},{"Start":"06:07.925 ","End":"06:11.270","Text":"Later, you\u0027ll be learning something called L\u0027Hopital\u0027s rule,"},{"Start":"06:11.270 ","End":"06:16.670","Text":"which is really magical and will make all this work happen in a matter of seconds."},{"Start":"06:16.670 ","End":"06:18.290","Text":"It will come later, but first,"},{"Start":"06:18.290 ","End":"06:20.890","Text":"you\u0027ll have to learn differentiation."},{"Start":"06:20.890 ","End":"06:22.590","Text":"Until such time,"},{"Start":"06:22.590 ","End":"06:23.930","Text":"you have to do it the hard way,"},{"Start":"06:23.930 ","End":"06:26.420","Text":"but you\u0027ll know the shortcut rule is on the way."},{"Start":"06:26.420 ","End":"06:29.880","Text":"That\u0027s all I have to say for this clip."}],"ID":8409},{"Watched":false,"Name":"Exercise 1","Duration":"3m 41s","ChapterTopicVideoID":1528,"CourseChapterTopicPlaylistID":163,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this exercise, we have to find the limit as x tends to 3."},{"Start":"00:04.230 ","End":"00:08.504","Text":"Now, it\u0027s an elementary function and if we\u0027re lucky,"},{"Start":"00:08.504 ","End":"00:10.500","Text":"and it\u0027s defined where x equals 3,"},{"Start":"00:10.500 ","End":"00:12.885","Text":"we just have to substitute x equals 3."},{"Start":"00:12.885 ","End":"00:15.210","Text":"Unfortunately, this is not the case."},{"Start":"00:15.210 ","End":"00:18.165","Text":"We put x equals 3 here,"},{"Start":"00:18.165 ","End":"00:25.725","Text":"we\u0027ll get that x squared minus 9 is 3 squared minus 9 is 0."},{"Start":"00:25.725 ","End":"00:27.390","Text":"A denominator of 0,"},{"Start":"00:27.390 ","End":"00:32.430","Text":"so 3 is not defined here so the trick of substitution won\u0027t work."},{"Start":"00:32.430 ","End":"00:34.315","Text":"We\u0027ll have to try something else."},{"Start":"00:34.315 ","End":"00:37.730","Text":"The standard trick to be used in cases like this"},{"Start":"00:37.730 ","End":"00:41.480","Text":"is to factorize both numerator and denominator."},{"Start":"00:41.480 ","End":"00:44.090","Text":"Let\u0027s start with the denominator."},{"Start":"00:44.090 ","End":"00:49.925","Text":"It\u0027s easier. We have x squared minus 9 equals."},{"Start":"00:49.925 ","End":"00:57.230","Text":"Now, let me remind you of a basic formula in algebra, difference of squares,"},{"Start":"00:57.230 ","End":"01:07.455","Text":"that a squared minus b squared is equal to a minus b, a plus b."},{"Start":"01:07.455 ","End":"01:09.795","Text":"If we apply it in our case,"},{"Start":"01:09.795 ","End":"01:11.745","Text":"9 is 3 squared."},{"Start":"01:11.745 ","End":"01:16.960","Text":"We get x minus 3, x plus 3."},{"Start":"01:16.960 ","End":"01:18.895","Text":"For the numerator,"},{"Start":"01:18.895 ","End":"01:23.915","Text":"I\u0027m going to have to give you another formula that when we have an expression,"},{"Start":"01:23.915 ","End":"01:29.630","Text":"x squared plus ax plus b,"},{"Start":"01:29.630 ","End":"01:39.365","Text":"this is equal to x minus x_1 times x minus x_2,"},{"Start":"01:39.365 ","End":"01:47.090","Text":"where x_1 and x_2 are the roots of the quadratic expression."},{"Start":"01:47.090 ","End":"01:51.260","Text":"In other words, the solutions to the equation where this is equal to 0."},{"Start":"01:51.260 ","End":"02:00.490","Text":"In our numerator, which is x squared minus x minus 6, to find the roots,"},{"Start":"02:00.490 ","End":"02:04.635","Text":"we solve the equation equals 0."},{"Start":"02:04.635 ","End":"02:08.690","Text":"Since you know how to solve the quadratic equations,"},{"Start":"02:08.690 ","End":"02:11.585","Text":"I\u0027ll just give you the answer straightaway."},{"Start":"02:11.585 ","End":"02:14.765","Text":"So applying what we wrote here,"},{"Start":"02:14.765 ","End":"02:21.740","Text":"we can say that x squared minus x minus 6 is"},{"Start":"02:21.740 ","End":"02:29.360","Text":"equal to x minus the first root times x minus the second root,"},{"Start":"02:29.360 ","End":"02:31.220","Text":"which makes it a plus,"},{"Start":"02:31.220 ","End":"02:34.205","Text":"and that factorizes this."},{"Start":"02:34.205 ","End":"02:36.335","Text":"Scroll down a bit."},{"Start":"02:36.335 ","End":"02:46.970","Text":"What we have if we write the original expression is the limit as x tends to 3."},{"Start":"02:46.970 ","End":"02:53.105","Text":"The numerator, which we factorized to be x minus 3,"},{"Start":"02:53.105 ","End":"02:56.900","Text":"x plus 2, and the denominator,"},{"Start":"02:56.900 ","End":"03:02.555","Text":"x minus 3 times x plus 3."},{"Start":"03:02.555 ","End":"03:05.615","Text":"Now, notice when x tends to 3,"},{"Start":"03:05.615 ","End":"03:07.580","Text":"it\u0027s not equal to 3."},{"Start":"03:07.580 ","End":"03:11.360","Text":"So x minus 3 is not 0 and we can cancel."},{"Start":"03:11.360 ","End":"03:16.550","Text":"At this point we have the limit as x goes to 3 of a new function,"},{"Start":"03:16.550 ","End":"03:19.085","Text":"x plus 2 over x plus 3."},{"Start":"03:19.085 ","End":"03:24.560","Text":"It\u0027s also elementary and here we can substitute x equals 3."},{"Start":"03:24.560 ","End":"03:27.290","Text":"So this is just equal to,"},{"Start":"03:27.290 ","End":"03:34.475","Text":"we get 3 plus 2/3 plus 3,"},{"Start":"03:34.475 ","End":"03:42.820","Text":"and that\u0027s just 5/6 and that\u0027s the answer. We\u0027re done."}],"ID":1540},{"Watched":false,"Name":"Exercise 2","Duration":"5m 5s","ChapterTopicVideoID":1529,"CourseChapterTopicPlaylistID":163,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.065","Text":"In this exercise, we have to find the limit as x goes to minus 5 of this function,"},{"Start":"00:07.065 ","End":"00:11.640","Text":"2x squared minus 50 over 2x squared plus 3x minus 35."},{"Start":"00:11.640 ","End":"00:13.860","Text":"This is an elementary function."},{"Start":"00:13.860 ","End":"00:18.345","Text":"If we\u0027re lucky enough that it\u0027s defined for minus 5,"},{"Start":"00:18.345 ","End":"00:21.505","Text":"then all we have to do is substitute minus 5."},{"Start":"00:21.505 ","End":"00:24.510","Text":"But unfortunately that\u0027s not so if you do"},{"Start":"00:24.510 ","End":"00:30.065","Text":"a mental computation of putting minus 5 for x in the denominator,"},{"Start":"00:30.065 ","End":"00:36.860","Text":"minus 5 squared is 25 times 2 is 50 minus 15 minus 35, we get a 0."},{"Start":"00:36.860 ","End":"00:40.460","Text":"In this case, we have to use the technique"},{"Start":"00:40.460 ","End":"00:44.285","Text":"of factorization of the numerator and the denominator."},{"Start":"00:44.285 ","End":"00:46.865","Text":"Let\u0027s start with the numerator."},{"Start":"00:46.865 ","End":"00:49.370","Text":"Here I wrote the formula I see we\u0027re going to need."},{"Start":"00:49.370 ","End":"00:59.390","Text":"Factorizing the numerator, we get 2x squared minus 50 is equal."},{"Start":"00:59.390 ","End":"01:02.030","Text":"First of all, let\u0027s take 2 outside the brackets,"},{"Start":"01:02.030 ","End":"01:10.440","Text":"2x squared minus 25 and 25 is 5 squared."},{"Start":"01:10.440 ","End":"01:12.200","Text":"If I use this formula,"},{"Start":"01:12.200 ","End":"01:14.420","Text":"which is the difference of squares formula,"},{"Start":"01:14.420 ","End":"01:23.505","Text":"we\u0027ll get that this is equal to 2 x minus 5, x plus 5."},{"Start":"01:23.505 ","End":"01:27.335","Text":"That\u0027s as far as the numerator can be simplified."},{"Start":"01:27.335 ","End":"01:29.095","Text":"Now the denominator."},{"Start":"01:29.095 ","End":"01:37.380","Text":"Here we have 2x squared plus 3x minus 35."},{"Start":"01:37.380 ","End":"01:41.170","Text":"I\u0027m going to need another formula for you."},{"Start":"01:41.170 ","End":"01:43.800","Text":"This is the formula I wanted to use,"},{"Start":"01:43.800 ","End":"01:46.130","Text":"that the quadratic expression,"},{"Start":"01:46.130 ","End":"01:50.995","Text":"x squared plus bx plus c is equal to a times x minus x_1,"},{"Start":"01:50.995 ","End":"01:52.890","Text":"x minus x_2,"},{"Start":"01:52.890 ","End":"01:58.070","Text":"and x_1 and x_2 are the roots of this quadratic expression,"},{"Start":"01:58.070 ","End":"02:02.650","Text":"which means the solution of the equation where all this is equal to 0."},{"Start":"02:02.650 ","End":"02:06.065","Text":"What we have to do first is find those roots."},{"Start":"02:06.065 ","End":"02:12.125","Text":"Let\u0027s just write here equal to 0 and we\u0027ll solve this equation."},{"Start":"02:12.125 ","End":"02:16.745","Text":"I\u0027m assuming that you know how to solve quadratic equations."},{"Start":"02:16.745 ","End":"02:18.965","Text":"I\u0027ll just tell you the answer."},{"Start":"02:18.965 ","End":"02:20.225","Text":"What this means,"},{"Start":"02:20.225 ","End":"02:22.025","Text":"according to this formula,"},{"Start":"02:22.025 ","End":"02:27.320","Text":"is that we can rewrite what was the denominator as 2x squared"},{"Start":"02:27.320 ","End":"02:35.210","Text":"plus 3x minus 35 is equal to 2,"},{"Start":"02:35.210 ","End":"02:37.865","Text":"that\u0027s the a part,"},{"Start":"02:37.865 ","End":"02:41.480","Text":"times x minus the first root,"},{"Start":"02:41.480 ","End":"02:43.820","Text":"7 over 2,"},{"Start":"02:43.820 ","End":"02:46.295","Text":"times x minus the other root,"},{"Start":"02:46.295 ","End":"02:50.165","Text":"x minus 5, let\u0027s just write it as plus 5."},{"Start":"02:50.165 ","End":"02:53.585","Text":"If I multiply the first brackets by 2,"},{"Start":"02:53.585 ","End":"02:55.835","Text":"it\u0027ll be a little bit simpler."},{"Start":"02:55.835 ","End":"02:58.835","Text":"I\u0027ll get 2x minus 7,"},{"Start":"02:58.835 ","End":"03:02.520","Text":"x plus 5 without fractions."},{"Start":"03:02.520 ","End":"03:07.640","Text":"Now what I\u0027m going to want to do is to substitute the numerator over"},{"Start":"03:07.640 ","End":"03:12.320","Text":"the denominator here in terms of the new factoring we found."},{"Start":"03:12.320 ","End":"03:18.620","Text":"I\u0027ll scroll down a bit and we can rewrite our exercise."},{"Start":"03:18.620 ","End":"03:23.895","Text":"The limit of 2x squared minus 50"},{"Start":"03:23.895 ","End":"03:31.485","Text":"over 2x squared plus 3x minus 35."},{"Start":"03:31.485 ","End":"03:34.310","Text":"Of course, I have to say where the limit is."},{"Start":"03:34.310 ","End":"03:38.450","Text":"It\u0027s where x tends to minus 5."},{"Start":"03:38.450 ","End":"03:43.670","Text":"This is equal to the limit of what we factored these as."},{"Start":"03:43.670 ","End":"03:51.980","Text":"The first 1, the numerator is twice x minus 5, x plus 5."},{"Start":"03:51.980 ","End":"03:55.475","Text":"The denominator, that\u0027s the work we did here,"},{"Start":"03:55.475 ","End":"04:01.085","Text":"is 2x minus 7 times x"},{"Start":"04:01.085 ","End":"04:07.725","Text":"plus 5 as x tends to minus 5."},{"Start":"04:07.725 ","End":"04:09.560","Text":"Now looking at this,"},{"Start":"04:09.560 ","End":"04:13.145","Text":"we have an x plus 5 in the numerator and in the denominator."},{"Start":"04:13.145 ","End":"04:15.590","Text":"Now, x is not equal to minus 5,"},{"Start":"04:15.590 ","End":"04:17.635","Text":"it only tends to minus 5,"},{"Start":"04:17.635 ","End":"04:19.485","Text":"so this thing is not 0."},{"Start":"04:19.485 ","End":"04:22.025","Text":"We can do this big cancellation."},{"Start":"04:22.025 ","End":"04:27.360","Text":"That really helps us because what remains is still an elementary expression,"},{"Start":"04:27.360 ","End":"04:31.545","Text":"only this time we can substitute minus 5 in it."},{"Start":"04:31.545 ","End":"04:34.460","Text":"The limit is just a substitution."},{"Start":"04:34.460 ","End":"04:37.640","Text":"We get twice minus 5,"},{"Start":"04:37.640 ","End":"04:46.720","Text":"minus 5 over twice minus 5, minus 7."},{"Start":"04:46.720 ","End":"04:49.940","Text":"The numerator is twice minus 10,"},{"Start":"04:49.940 ","End":"04:52.070","Text":"which is minus 20."},{"Start":"04:52.070 ","End":"04:57.650","Text":"The denominator minus 10 minus 7 is minus 17."},{"Start":"04:57.650 ","End":"05:03.020","Text":"The answer is 20 over 17."},{"Start":"05:03.020 ","End":"05:06.490","Text":"This is it and we\u0027re done."}],"ID":1541},{"Watched":false,"Name":"Exercise 3","Duration":"3m 8s","ChapterTopicVideoID":1530,"CourseChapterTopicPlaylistID":163,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we have to find the limit when x tends to"},{"Start":"00:04.170 ","End":"00:10.020","Text":"1 of the function x^7 minus x over x minus 1."},{"Start":"00:10.020 ","End":"00:12.255","Text":"It\u0027s an elementary function."},{"Start":"00:12.255 ","End":"00:14.905","Text":"If we could substitute x equals 1,"},{"Start":"00:14.905 ","End":"00:16.310","Text":"that would give us the answer."},{"Start":"00:16.310 ","End":"00:19.220","Text":"Unfortunately, we can\u0027t substitute x equals 1"},{"Start":"00:19.220 ","End":"00:22.730","Text":"because of the denominator that would make it 0."},{"Start":"00:22.730 ","End":"00:25.790","Text":"We have to use the technique of factorization."},{"Start":"00:25.790 ","End":"00:28.250","Text":"We factorize a numerator and denominator,"},{"Start":"00:28.250 ","End":"00:30.400","Text":"and hopefully something cancels."},{"Start":"00:30.400 ","End":"00:32.700","Text":"The denominator doesn\u0027t need to be factored."},{"Start":"00:32.700 ","End":"00:40.400","Text":"Let\u0027s factorize the numerator and let\u0027s see what is x^7 minus x, what this equals."},{"Start":"00:40.400 ","End":"00:42.410","Text":"I have to show you a formula."},{"Start":"00:42.410 ","End":"00:45.395","Text":"Here is the formula and I\u0027ll return to that when I need it."},{"Start":"00:45.395 ","End":"00:48.820","Text":"In our case, we first factor out an x."},{"Start":"00:48.820 ","End":"00:55.900","Text":"We get x, x times x^6 minus 1."},{"Start":"00:55.900 ","End":"01:01.520","Text":"Now, I\u0027m going to use the formula over here,"},{"Start":"01:01.520 ","End":"01:05.990","Text":"this time with n equals 6."},{"Start":"01:05.990 ","End":"01:09.190","Text":"This will give us that if this is 6,"},{"Start":"01:09.190 ","End":"01:15.520","Text":"here we\u0027ll have a^5th plus a^4,"},{"Start":"01:15.520 ","End":"01:18.890","Text":"plus a^3 like cubed,"},{"Start":"01:18.890 ","End":"01:24.455","Text":"plus a squared plus a plus 1."},{"Start":"01:24.455 ","End":"01:26.660","Text":"That\u0027s just for this bit here,"},{"Start":"01:26.660 ","End":"01:30.755","Text":"cause we still have the a minus 1."},{"Start":"01:30.755 ","End":"01:35.160","Text":"In our case, x is the role of a,"},{"Start":"01:35.160 ","End":"01:45.100","Text":"so we get this thing equals x times the x minus 1 times this thing,"},{"Start":"01:45.100 ","End":"01:51.695","Text":"but with x instead of a. X^5, plus x^4,"},{"Start":"01:51.695 ","End":"01:54.075","Text":"plus x cubed,"},{"Start":"01:54.075 ","End":"01:56.310","Text":"plus x squared,"},{"Start":"01:56.310 ","End":"02:00.075","Text":"plus x plus 1."},{"Start":"02:00.075 ","End":"02:04.560","Text":"Now, we have to rewrite this limit."},{"Start":"02:04.560 ","End":"02:15.025","Text":"What we\u0027re looking for is the limit as x tends to or goes to 1 of x^7 minus x,"},{"Start":"02:15.025 ","End":"02:16.075","Text":"which is all this."},{"Start":"02:16.075 ","End":"02:20.035","Text":"I\u0027ll just copy it out again quickly,"},{"Start":"02:20.035 ","End":"02:25.430","Text":"over the original x minus 1."},{"Start":"02:25.430 ","End":"02:27.770","Text":"Now, here we\u0027re lucky."},{"Start":"02:27.770 ","End":"02:30.725","Text":"We can cancel the x minus 1."},{"Start":"02:30.725 ","End":"02:34.670","Text":"Notice that this is not 0 because when x tends to 1,"},{"Start":"02:34.670 ","End":"02:36.410","Text":"it\u0027s not equal to 1."},{"Start":"02:36.410 ","End":"02:41.060","Text":"What\u0027s left is a polynomial an elementary function."},{"Start":"02:41.060 ","End":"02:45.620","Text":"To find the limit we just have to substitute 1 instead of x."},{"Start":"02:45.620 ","End":"02:53.960","Text":"This is equal to 1 times 1^5, plus 1^4,"},{"Start":"02:53.960 ","End":"02:57.265","Text":"plus 1 cubed,"},{"Start":"02:57.265 ","End":"03:04.665","Text":"plus 1 squared plus 1 and another 1 is 6."},{"Start":"03:04.665 ","End":"03:09.850","Text":"So that\u0027s the answer to the exercise. We\u0027re done."}],"ID":1542},{"Watched":false,"Name":"Exercise 4","Duration":"4m 21s","ChapterTopicVideoID":1531,"CourseChapterTopicPlaylistID":163,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we have to find the limit as x tends to"},{"Start":"00:03.780 ","End":"00:09.990","Text":"1 of x to the n minus x over x minus 1."},{"Start":"00:09.990 ","End":"00:14.550","Text":"I should add that we have to assume that n is some positive whole number,"},{"Start":"00:14.550 ","End":"00:19.470","Text":"preferably bigger than 1 and integer."},{"Start":"00:19.470 ","End":"00:25.800","Text":"This function x to the n minus x over x minus 1 is an elementary function."},{"Start":"00:25.800 ","End":"00:29.505","Text":"If we could substitute x equals 1,"},{"Start":"00:29.505 ","End":"00:30.705","Text":"if that were allowed,"},{"Start":"00:30.705 ","End":"00:32.955","Text":"that would just give us our answer."},{"Start":"00:32.955 ","End":"00:37.170","Text":"But we can\u0027t because if we put x equals 1 in the denominator,"},{"Start":"00:37.170 ","End":"00:38.820","Text":"we see that we get 0."},{"Start":"00:38.820 ","End":"00:45.649","Text":"We\u0027ll have to use a technique of factoring numerator and denominator. Let\u0027s begin."},{"Start":"00:45.649 ","End":"00:55.069","Text":"The limit as x tends to 1 of x to the n minus x"},{"Start":"00:55.069 ","End":"01:05.505","Text":"over x minus 1 is equal to the limit as x goes to 1."},{"Start":"01:05.505 ","End":"01:09.140","Text":"We can take x out as a factor from the numerator,"},{"Start":"01:09.140 ","End":"01:20.070","Text":"x times x to the n minus 1 minus 1 over x minus 1."},{"Start":"01:20.070 ","End":"01:23.750","Text":"We have to know what to do with this term here."},{"Start":"01:23.750 ","End":"01:27.120","Text":"Let me remind you of a formula."},{"Start":"01:27.170 ","End":"01:29.995","Text":"This is the formula I mean."},{"Start":"01:29.995 ","End":"01:35.150","Text":"I\u0027ve deliberately chosen other letters than x and n to make it general,"},{"Start":"01:35.150 ","End":"01:37.100","Text":"that a to the power of k,"},{"Start":"01:37.100 ","End":"01:38.615","Text":"where k is a natural number,"},{"Start":"01:38.615 ","End":"01:44.900","Text":"minus 1 is a minus 1 times the sum of descending powers of a,"},{"Start":"01:44.900 ","End":"01:46.895","Text":"a to the k minus 1,"},{"Start":"01:46.895 ","End":"01:51.140","Text":"a to the k minus 2 and so on plus a plus 1."},{"Start":"01:51.140 ","End":"01:54.890","Text":"If we apply that here, continuing,"},{"Start":"01:54.890 ","End":"01:58.775","Text":"we get that this is equal to"},{"Start":"01:58.775 ","End":"02:05.930","Text":"the limit as x goes to 1 of x in the numerator."},{"Start":"02:05.930 ","End":"02:12.690","Text":"Now, this thing, it\u0027s like k is n minus 1 and a is x."},{"Start":"02:12.860 ","End":"02:17.280","Text":"The a minus 1 is the x minus 1."},{"Start":"02:17.280 ","End":"02:20.635","Text":"This sum begins from k minus 1."},{"Start":"02:20.635 ","End":"02:24.310","Text":"In other words, it begins from n minus 2,"},{"Start":"02:24.310 ","End":"02:34.600","Text":"so it\u0027s x to the n minus 2 plus x to the n minus 3 and so on plus x plus 1."},{"Start":"02:34.600 ","End":"02:40.445","Text":"The denominator is the same old denominator x minus 1."},{"Start":"02:40.445 ","End":"02:46.610","Text":"But the x minus 1 is in the numerator and in the denominator,"},{"Start":"02:46.610 ","End":"02:49.180","Text":"so we can cancel it."},{"Start":"02:49.180 ","End":"02:53.700","Text":"It\u0027s not 0 because when x tends to 1, it isn\u0027t 1."},{"Start":"02:53.700 ","End":"03:00.045","Text":"What we\u0027re left with is the limit as x goes to 1 of"},{"Start":"03:00.045 ","End":"03:07.245","Text":"x times x to the n minus 2 plus and so on,"},{"Start":"03:07.245 ","End":"03:12.585","Text":"plus x plus 1 over 1."},{"Start":"03:12.585 ","End":"03:18.050","Text":"This equals, all I have to do is substitute instead of x, 1,"},{"Start":"03:18.050 ","End":"03:24.205","Text":"and I get 1 times 1 plus dot dot dot,"},{"Start":"03:24.205 ","End":"03:27.390","Text":"plus 1, plus 1."},{"Start":"03:27.390 ","End":"03:31.880","Text":"Now, the question is how many 1s are there?"},{"Start":"03:31.880 ","End":"03:34.045","Text":"Well, if you notice,"},{"Start":"03:34.045 ","End":"03:38.335","Text":"we begin with this 1 which is x to the power of 0,"},{"Start":"03:38.335 ","End":"03:40.075","Text":"then x to the power of 1."},{"Start":"03:40.075 ","End":"03:42.490","Text":"In other words, the powers go,"},{"Start":"03:42.490 ","End":"03:44.710","Text":"if I take it from left to right,"},{"Start":"03:44.710 ","End":"03:46.930","Text":"the powers are 0,"},{"Start":"03:46.930 ","End":"03:52.005","Text":"1 and so on up to n minus 2."},{"Start":"03:52.005 ","End":"03:57.085","Text":"Now, if I take all the numbers from 0 up to a certain number,"},{"Start":"03:57.085 ","End":"04:01.960","Text":"then there\u0027s going to be a total of n minus 1 of them."},{"Start":"04:01.960 ","End":"04:03.910","Text":"If it was from 0 to,"},{"Start":"04:03.910 ","End":"04:05.425","Text":"let\u0027s say, 8,"},{"Start":"04:05.425 ","End":"04:07.885","Text":"there\u0027d be 9 of them and so on."},{"Start":"04:07.885 ","End":"04:12.075","Text":"What we have is n minus 1, 1s."},{"Start":"04:12.075 ","End":"04:15.435","Text":"1 plus 1 plus 1 plus 1 n minus 1 times."},{"Start":"04:15.435 ","End":"04:22.180","Text":"That just gives us n minus 1 and that\u0027s our answer. We\u0027re done."}],"ID":1543}],"Thumbnail":null,"ID":163},{"Name":"Technique 3 Multiplying by the Conjugate","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Multiplying by The Conjugate","Duration":"8m 13s","ChapterTopicVideoID":8250,"CourseChapterTopicPlaylistID":164,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this clip, we\u0027re going to talk about technique number 3,"},{"Start":"00:03.900 ","End":"00:08.910","Text":"and it\u0027s called multiplying by the conjugate or multiplication with the conjugate."},{"Start":"00:08.910 ","End":"00:12.555","Text":"First, I\u0027ll tell you what the indications that we should use it."},{"Start":"00:12.555 ","End":"00:14.655","Text":"Well, there\u0027s 2 main things."},{"Start":"00:14.655 ","End":"00:19.070","Text":"1 is that the limit is of the form 0/0."},{"Start":"00:19.070 ","End":"00:20.600","Text":"It means when you substitute the value,"},{"Start":"00:20.600 ","End":"00:23.285","Text":"that\u0027s what it is, that\u0027s what you get."},{"Start":"00:23.285 ","End":"00:27.060","Text":"The second thing is that there"},{"Start":"00:27.060 ","End":"00:37.065","Text":"is a square root sign,"},{"Start":"00:37.065 ","End":"00:39.380","Text":"could be numerator or denominator."},{"Start":"00:39.380 ","End":"00:40.985","Text":"Let\u0027s start with an example."},{"Start":"00:40.985 ","End":"00:46.550","Text":"The example will be limit as x goes to 1"},{"Start":"00:46.550 ","End":"00:53.360","Text":"of the square root of 15 plus x minus 4,"},{"Start":"00:53.360 ","End":"00:58.305","Text":"all over x minus 1."},{"Start":"00:58.305 ","End":"01:01.625","Text":"Let\u0027s see, do these conditions hold?"},{"Start":"01:01.625 ","End":"01:03.470","Text":"Well, if I put x as 1,"},{"Start":"01:03.470 ","End":"01:06.095","Text":"15 plus x is 16, square root of 16,"},{"Start":"01:06.095 ","End":"01:08.740","Text":"4 minus 4 is 0,"},{"Start":"01:08.740 ","End":"01:11.700","Text":"and 1 minus 1 is 0."},{"Start":"01:11.700 ","End":"01:18.500","Text":"We do indeed have the 0/0 form."},{"Start":"01:18.500 ","End":"01:23.330","Text":"Secondly, you want to be looking at the numerator and/or the denominator,"},{"Start":"01:23.330 ","End":"01:25.055","Text":"and we do have a square root."},{"Start":"01:25.055 ","End":"01:27.520","Text":"There is the square root."},{"Start":"01:27.980 ","End":"01:30.765","Text":"Both of these conditions hold,"},{"Start":"01:30.765 ","End":"01:35.600","Text":"so what is a conjugate and what do I mean by multiplying by the conjugate?"},{"Start":"01:35.600 ","End":"01:37.310","Text":"Well, I\u0027ll show you."},{"Start":"01:37.310 ","End":"01:39.290","Text":"If we have a difference of 2 terms,"},{"Start":"01:39.290 ","End":"01:40.730","Text":"A minus B,"},{"Start":"01:40.730 ","End":"01:45.320","Text":"then its conjugate is A plus B and vice versa."},{"Start":"01:45.320 ","End":"01:50.030","Text":"If we multiply an expression by its conjugate,"},{"Start":"01:50.030 ","End":"01:53.965","Text":"what we get is A minus B times A plus B."},{"Start":"01:53.965 ","End":"01:58.550","Text":"There\u0027s a formula that it\u0027s equal to A squared minus B squared."},{"Start":"01:58.550 ","End":"02:00.410","Text":"What is this good for?"},{"Start":"02:00.410 ","End":"02:05.675","Text":"Well, if A and/or B has a square root over it,"},{"Start":"02:05.675 ","End":"02:07.070","Text":"then when we square it,"},{"Start":"02:07.070 ","End":"02:09.380","Text":"we\u0027ll get rid of the square root."},{"Start":"02:09.380 ","End":"02:14.520","Text":"Let\u0027s see what happens in our example."},{"Start":"02:14.900 ","End":"02:22.915","Text":"I\u0027m taking the example of A and B from the numerator of our example here."},{"Start":"02:22.915 ","End":"02:25.000","Text":"Let\u0027s just, as a side exercise,"},{"Start":"02:25.000 ","End":"02:29.885","Text":"take that numerator and let\u0027s multiply it by its conjugate."},{"Start":"02:29.885 ","End":"02:32.050","Text":"Instead of the minus here,"},{"Start":"02:32.050 ","End":"02:33.515","Text":"I have a plus here,"},{"Start":"02:33.515 ","End":"02:38.180","Text":"and if we multiply them using this formula,"},{"Start":"02:38.700 ","End":"02:44.905","Text":"what we get is the square root"},{"Start":"02:44.905 ","End":"02:52.515","Text":"of 15 plus x squared,"},{"Start":"02:52.515 ","End":"02:58.695","Text":"that\u0027s my A squared minus B squared,"},{"Start":"02:58.695 ","End":"03:02.050","Text":"which is 4 squared."},{"Start":"03:02.300 ","End":"03:09.830","Text":"Now, taking the square of the square root just leaves us with the expression itself,"},{"Start":"03:09.830 ","End":"03:12.510","Text":"which is 15 plus x."},{"Start":"03:13.940 ","End":"03:19.490","Text":"Here, that\u0027s the essence of the multiplication with the conjugate,"},{"Start":"03:19.490 ","End":"03:22.610","Text":"is that if you take something can multiply by its conjugate,"},{"Start":"03:22.610 ","End":"03:24.790","Text":"you don\u0027t have any square roots anymore."},{"Start":"03:24.790 ","End":"03:28.170","Text":"It\u0027s 15 plus x minus this 4 squared,"},{"Start":"03:28.170 ","End":"03:36.180","Text":"which is 16, which ultimately just leaves us with x minus 1."},{"Start":"03:36.880 ","End":"03:39.080","Text":"Now, having done this,"},{"Start":"03:39.080 ","End":"03:41.405","Text":"we return to our example."},{"Start":"03:41.405 ","End":"03:45.020","Text":"How we would solve it is that we would take"},{"Start":"03:45.020 ","End":"03:51.910","Text":"this expression and write it in a slightly different form algebraically."},{"Start":"03:51.910 ","End":"03:55.780","Text":"X goes to 1 of,"},{"Start":"03:55.780 ","End":"03:58.655","Text":"first of all, I\u0027m going to copy the same expression,"},{"Start":"03:58.655 ","End":"04:06.440","Text":"15 plus x minus 4 over x minus 1."},{"Start":"04:06.440 ","End":"04:10.640","Text":"I would like to multiply this by its conjugate,"},{"Start":"04:10.640 ","End":"04:18.810","Text":"which is the square root of 15 plus x plus 4."},{"Start":"04:18.810 ","End":"04:21.920","Text":"But of course, you can\u0027t just multiply it by"},{"Start":"04:21.920 ","End":"04:24.619","Text":"something because it won\u0027t be the same exercise."},{"Start":"04:24.619 ","End":"04:32.975","Text":"I have to compensate and multiply the numerator or the denominator also by same thing,"},{"Start":"04:32.975 ","End":"04:38.810","Text":"square root of 15 plus x plus 4, same thing."},{"Start":"04:38.810 ","End":"04:42.800","Text":"Something over itself is just 1 and you can multiply by 1,"},{"Start":"04:42.800 ","End":"04:45.185","Text":"leave the expression unchanged."},{"Start":"04:45.185 ","End":"04:56.115","Text":"But if we do a little bit of algebra here and multiply it out,"},{"Start":"04:56.115 ","End":"04:59.060","Text":"well, this times this, we\u0027ve already done."},{"Start":"04:59.060 ","End":"05:00.860","Text":"In fact, we\u0027ve done it over here."},{"Start":"05:00.860 ","End":"05:03.035","Text":"No need to do it again."},{"Start":"05:03.035 ","End":"05:06.545","Text":"This equals the limit of"},{"Start":"05:06.545 ","End":"05:11.675","Text":"this times this is x minus 1 is all we have left on the numerator,"},{"Start":"05:11.675 ","End":"05:12.950","Text":"and on the denominator,"},{"Start":"05:12.950 ","End":"05:15.050","Text":"we have this thing times this thing,"},{"Start":"05:15.050 ","End":"05:20.290","Text":"so we have, just put it in brackets first."},{"Start":"05:20.290 ","End":"05:22.860","Text":"We have from here,"},{"Start":"05:22.860 ","End":"05:26.370","Text":"the x minus 1, and from here,"},{"Start":"05:26.370 ","End":"05:30.000","Text":"the square root of"},{"Start":"05:30.000 ","End":"05:39.240","Text":"15 plus x plus 4."},{"Start":"05:39.240 ","End":"05:43.955","Text":"Now, this is where we come to familiar territory."},{"Start":"05:43.955 ","End":"05:48.720","Text":"We can now use the factorizing canceling, in this case,"},{"Start":"05:48.720 ","End":"05:51.740","Text":"canceling technique, and just look x minus 1 and x"},{"Start":"05:51.740 ","End":"05:55.190","Text":"minus 1 because I have to leave a 1 here."},{"Start":"05:55.190 ","End":"05:57.620","Text":"It\u0027s going to be something on the numerator."},{"Start":"05:57.620 ","End":"06:00.245","Text":"Now in this expression,"},{"Start":"06:00.245 ","End":"06:02.570","Text":"there are no problems,"},{"Start":"06:02.570 ","End":"06:04.325","Text":"there\u0027s no zeros in the denominator."},{"Start":"06:04.325 ","End":"06:06.290","Text":"We could just use substitution."},{"Start":"06:06.290 ","End":"06:11.460","Text":"I\u0027m going to substitute x equals 1 in here,"},{"Start":"06:11.650 ","End":"06:20.670","Text":"and what we will get is 1 over,"},{"Start":"06:20.670 ","End":"06:22.635","Text":"let\u0027s see if we can do this in our heads,"},{"Start":"06:22.635 ","End":"06:26.145","Text":"15 plus 1 is 16,"},{"Start":"06:26.145 ","End":"06:30.780","Text":"square root of 16 is 4 plus 4 just comes out as 8,"},{"Start":"06:30.780 ","End":"06:35.370","Text":"so 1/8, that\u0027s the answer."},{"Start":"06:35.370 ","End":"06:37.905","Text":"Not too bad. A little bit of algebra."},{"Start":"06:37.905 ","End":"06:40.200","Text":"Be careful with those square roots."},{"Start":"06:40.200 ","End":"06:45.870","Text":"Mainly, to multiply by the conjugate and then we get rid of square roots."},{"Start":"06:49.510 ","End":"06:56.990","Text":"The essential step in this exercise I\u0027d like to point out was this product."},{"Start":"06:56.990 ","End":"06:59.765","Text":"We multiply by something over itself,"},{"Start":"06:59.765 ","End":"07:01.970","Text":"and 1 of these is the conjugate,"},{"Start":"07:01.970 ","End":"07:03.740","Text":"either the numerator or the denominator is"},{"Start":"07:03.740 ","End":"07:07.205","Text":"the conjugate of what we had in the original exercise."},{"Start":"07:07.205 ","End":"07:10.470","Text":"It\u0027s not too bad, but it\u0027s still fairly difficult,"},{"Start":"07:10.470 ","End":"07:14.075","Text":"you\u0027ll have plenty more such exercises to practice on,"},{"Start":"07:14.075 ","End":"07:17.675","Text":"but I would like to say something important,"},{"Start":"07:17.675 ","End":"07:20.280","Text":"that in the future,"},{"Start":"07:20.450 ","End":"07:23.190","Text":"these things will get a lot easier,"},{"Start":"07:23.190 ","End":"07:25.790","Text":"when you\u0027ve learned L\u0027Hopital\u0027s Rule,"},{"Start":"07:25.790 ","End":"07:27.665","Text":"I\u0027ll just write his name."},{"Start":"07:27.665 ","End":"07:29.280","Text":"He\u0027s a French,"},{"Start":"07:29.280 ","End":"07:31.670","Text":"I\u0027m not sure in which century."},{"Start":"07:31.670 ","End":"07:37.220","Text":"But there was a clever fellow called L\u0027Hopital,"},{"Start":"07:37.220 ","End":"07:41.820","Text":"and he had rules for limits,"},{"Start":"07:41.820 ","End":"07:46.070","Text":"mainly of the form 0/0 or infinity over infinity,"},{"Start":"07:46.070 ","End":"07:47.780","Text":"and once we\u0027ve learned these,"},{"Start":"07:47.780 ","End":"07:49.520","Text":"things will get a whole lot easier."},{"Start":"07:49.520 ","End":"07:55.640","Text":"Thing is that you have to know differentiation and the use of derivatives,"},{"Start":"07:55.640 ","End":"07:57.665","Text":"and once you\u0027ve learned those,"},{"Start":"07:57.665 ","End":"08:01.895","Text":"we can learn L\u0027Hopital\u0027s Rule, and then things will get a whole lot easier."},{"Start":"08:01.895 ","End":"08:06.095","Text":"Just to keep you encouraged or don\u0027t get discouraged."},{"Start":"08:06.095 ","End":"08:12.870","Text":"Anyway, that\u0027s it for multiplying by the conjugate. We\u0027re done."}],"ID":8410},{"Watched":false,"Name":"Exercise 1","Duration":"4m 3s","ChapterTopicVideoID":1532,"CourseChapterTopicPlaylistID":164,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this exercise, we have to find the limit as x tends to 1 of the function,"},{"Start":"00:05.250 ","End":"00:08.955","Text":"1 minus the square root of x over 1 minus x."},{"Start":"00:08.955 ","End":"00:11.265","Text":"This is an elementary function."},{"Start":"00:11.265 ","End":"00:16.365","Text":"If we\u0027re lucky, we can just substitute x equals 1 and get the answer."},{"Start":"00:16.365 ","End":"00:19.950","Text":"Unfortunately, we\u0027re not because if you substitute x"},{"Start":"00:19.950 ","End":"00:23.525","Text":"equals 1 in the denominator, you get 0."},{"Start":"00:23.525 ","End":"00:25.860","Text":"In the numerator, square root of 1 is 1,"},{"Start":"00:25.860 ","End":"00:28.215","Text":"1 minus 1 is also 0."},{"Start":"00:28.215 ","End":"00:34.560","Text":"We have a limit of the form 0 over 0 and we have to apply some technique."},{"Start":"00:34.560 ","End":"00:39.780","Text":"Looking at it, we see that 1 of the terms has a square root over it."},{"Start":"00:39.780 ","End":"00:43.670","Text":"This indicates using the method of the conjugate."},{"Start":"00:43.670 ","End":"00:45.845","Text":"What is a conjugate in general?"},{"Start":"00:45.845 ","End":"00:49.460","Text":"In general, the conjugate of"},{"Start":"00:49.460 ","End":"00:56.309","Text":"a plus b is a minus b and vice versa."},{"Start":"00:56.309 ","End":"00:57.965","Text":"Each conjugates of the other."},{"Start":"00:57.965 ","End":"01:01.850","Text":"Interesting thing, or the useful thing about conjugates is that if you"},{"Start":"01:01.850 ","End":"01:07.330","Text":"multiply them a plus b times a minus b,"},{"Start":"01:07.330 ","End":"01:09.965","Text":"using the difference of squares formula,"},{"Start":"01:09.965 ","End":"01:12.950","Text":"we get a squared minus b squared."},{"Start":"01:12.950 ","End":"01:15.500","Text":"If a or b is a square root sign,"},{"Start":"01:15.500 ","End":"01:19.675","Text":"after expanding here, the square root sign will disappear."},{"Start":"01:19.675 ","End":"01:22.880","Text":"This would be useful for us in our case."},{"Start":"01:22.880 ","End":"01:25.970","Text":"Now, let\u0027s take a look at our numerator."},{"Start":"01:25.970 ","End":"01:30.935","Text":"If we take this as,1 as a and square root of x as b,"},{"Start":"01:30.935 ","End":"01:35.150","Text":"then the conjugate would be 1 plus the square root of x."},{"Start":"01:35.150 ","End":"01:38.255","Text":"If we multiply, we\u0027ll get rid of the square roots."},{"Start":"01:38.255 ","End":"01:41.410","Text":"Let\u0027s just play with the numerator at first."},{"Start":"01:41.410 ","End":"01:45.950","Text":"If we see what is 1 minus the square root of"},{"Start":"01:45.950 ","End":"01:51.410","Text":"x and multiply it by its conjugate 1 plus the square root of x,"},{"Start":"01:51.410 ","End":"01:56.480","Text":"we\u0027ll get a squared minus b squared is 1 squared,"},{"Start":"01:56.480 ","End":"02:01.520","Text":"which is 1 minus the square root of x squared is just x."},{"Start":"02:01.520 ","End":"02:04.610","Text":"This will help us to write this fraction in"},{"Start":"02:04.610 ","End":"02:08.360","Text":"a different form where it will be easier to find the limit."},{"Start":"02:08.360 ","End":"02:16.910","Text":"Let\u0027s see, the limit as x tends to 1 of 1 minus the square root of"},{"Start":"02:16.910 ","End":"02:25.460","Text":"x over 1 minus x will equal the limit as x goes to 1."},{"Start":"02:25.460 ","End":"02:31.490","Text":"Now what I want to do is multiply the numerator by the conjugate 1 plus square root of x."},{"Start":"02:31.490 ","End":"02:34.580","Text":"But I can\u0027t just multiply the numerator."},{"Start":"02:34.580 ","End":"02:39.080","Text":"What I\u0027d like to do then is to multiply this by 1"},{"Start":"02:39.080 ","End":"02:43.190","Text":"plus the square root of x and that will simplify it."},{"Start":"02:43.190 ","End":"02:46.609","Text":"But as I say, you can\u0027t just multiply the numerator of a fraction."},{"Start":"02:46.609 ","End":"02:49.370","Text":"If I multiply the denominator also,"},{"Start":"02:49.370 ","End":"02:55.610","Text":"then what I\u0027m really doing is I\u0027m multiplying the fraction by 1,"},{"Start":"02:55.610 ","End":"02:57.950","Text":"so I\u0027m not changing it in any way."},{"Start":"02:57.950 ","End":"03:04.520","Text":"If we continue, we get the limit x goes to 1 of."},{"Start":"03:04.520 ","End":"03:12.725","Text":"Now this times this we\u0027ve already done as a side exercise over here and that\u0027s 1 minus x."},{"Start":"03:12.725 ","End":"03:16.370","Text":"The denominator is this times this,"},{"Start":"03:16.370 ","End":"03:24.014","Text":"so it\u0027s 1 minus x times 1 plus the square root x."},{"Start":"03:24.014 ","End":"03:29.280","Text":"Here we are lucky that 1 minus x cancels."},{"Start":"03:29.280 ","End":"03:32.690","Text":"In the numerator, we\u0027re left with just 1."},{"Start":"03:32.690 ","End":"03:36.200","Text":"We can cancel because x tends to 1,"},{"Start":"03:36.200 ","End":"03:37.865","Text":"but it is not equal to 1."},{"Start":"03:37.865 ","End":"03:42.845","Text":"Now all we\u0027re left with is 1 over 1 plus the square root of x."},{"Start":"03:42.845 ","End":"03:50.390","Text":"This is also an elementary function and this time we can substitute x equals 1 here."},{"Start":"03:50.390 ","End":"03:58.475","Text":"We get that this equals 1 over 1 plus the square root of 1."},{"Start":"03:58.475 ","End":"04:04.530","Text":"Obviously this is equal to 1/2 and that\u0027s our answer."}],"ID":1544},{"Watched":false,"Name":"Exercise 2","Duration":"4m 29s","ChapterTopicVideoID":1533,"CourseChapterTopicPlaylistID":164,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we have to find the limit as x tends to 3"},{"Start":"00:04.500 ","End":"00:10.980","Text":"of this function of x minus 3 over the square root of x plus 1 minus 2."},{"Start":"00:10.980 ","End":"00:12.720","Text":"It\u0027s an elementary function,"},{"Start":"00:12.720 ","End":"00:14.280","Text":"and if we\u0027re lucky,"},{"Start":"00:14.280 ","End":"00:17.910","Text":"we could just substitute x equals 3 and get the answer."},{"Start":"00:17.910 ","End":"00:19.935","Text":"But this is not so."},{"Start":"00:19.935 ","End":"00:23.160","Text":"Because if we put x equals 3 in the numerator,"},{"Start":"00:23.160 ","End":"00:24.540","Text":"we get 3 minus 3,"},{"Start":"00:24.540 ","End":"00:26.370","Text":"which is 0,"},{"Start":"00:26.370 ","End":"00:31.025","Text":"and if we put x equals 3 in the denominator, is also 0."},{"Start":"00:31.025 ","End":"00:34.190","Text":"So we have an expression of the form 0 over 0,"},{"Start":"00:34.190 ","End":"00:36.905","Text":"and we have to use some technique."},{"Start":"00:36.905 ","End":"00:40.655","Text":"I notice that there is a square root here,"},{"Start":"00:40.655 ","End":"00:43.550","Text":"and if 1 of the terms or more in"},{"Start":"00:43.550 ","End":"00:46.640","Text":"the numerator or denominator is under a square root sign,"},{"Start":"00:46.640 ","End":"00:50.330","Text":"that indicates that we should probably be using conjugates."},{"Start":"00:50.330 ","End":"00:52.115","Text":"What is a conjugate?"},{"Start":"00:52.115 ","End":"00:57.680","Text":"Well, in general, the conjugate of an expression A plus B"},{"Start":"00:57.680 ","End":"01:03.780","Text":"is just simply the opposite sign A minus B,"},{"Start":"01:03.780 ","End":"01:08.810","Text":"and conversely, the conjugate of A minus B is A plus B."},{"Start":"01:08.810 ","End":"01:13.885","Text":"This is often useful because when you multiply 2 conjugates,"},{"Start":"01:13.885 ","End":"01:18.375","Text":"in this case, A plus B times A minus B,"},{"Start":"01:18.375 ","End":"01:25.775","Text":"we get A squared minus B squared using the famous difference of squares formula."},{"Start":"01:25.775 ","End":"01:28.760","Text":"So if A or B was a square root,"},{"Start":"01:28.760 ","End":"01:30.470","Text":"after we square that,"},{"Start":"01:30.470 ","End":"01:32.150","Text":"it will no longer be a square root,"},{"Start":"01:32.150 ","End":"01:34.735","Text":"and that usually helps."},{"Start":"01:34.735 ","End":"01:38.225","Text":"In our case, here\u0027s the square root."},{"Start":"01:38.225 ","End":"01:42.155","Text":"I proposed to let the square root of x plus 1 be A,"},{"Start":"01:42.155 ","End":"01:45.500","Text":"and 2 can be B, and we\u0027ll have A minus B."},{"Start":"01:45.500 ","End":"01:47.539","Text":"Let\u0027s first of all, as an exercise,"},{"Start":"01:47.539 ","End":"01:51.520","Text":"see what happens if we multiply the denominator by its conjugate?"},{"Start":"01:51.520 ","End":"01:57.080","Text":"A square root of x plus 1 minus 2,"},{"Start":"01:57.080 ","End":"01:59.225","Text":"and I\u0027ll use brackets here."},{"Start":"01:59.225 ","End":"02:01.580","Text":"I\u0027ll multiply it by its conjugate,"},{"Start":"02:01.580 ","End":"02:06.305","Text":"which is the square root of x plus 1 plus 2."},{"Start":"02:06.305 ","End":"02:10.000","Text":"This equals, using this formula,"},{"Start":"02:10.000 ","End":"02:12.170","Text":"A squared minus B squared."},{"Start":"02:12.170 ","End":"02:19.160","Text":"The square root of x plus 1 squared is just x plus 1 without the square root,"},{"Start":"02:19.160 ","End":"02:22.685","Text":"and then minus the 2 squared."},{"Start":"02:22.685 ","End":"02:28.160","Text":"In short, this is just equal to x minus 3."},{"Start":"02:28.160 ","End":"02:31.280","Text":"This is going to help us to solve this limit."},{"Start":"02:31.280 ","End":"02:32.855","Text":"Let\u0027s continue here."},{"Start":"02:32.855 ","End":"02:40.640","Text":"The limit as x tends to 3 of x minus 3"},{"Start":"02:40.640 ","End":"02:47.135","Text":"over square root of x plus 1 minus 2."},{"Start":"02:47.135 ","End":"02:55.850","Text":"What we can do is to multiply top and bottom by the conjugate of the denominator."},{"Start":"02:55.850 ","End":"02:58.895","Text":"I\u0027d like to multiply this by its conjugate,"},{"Start":"02:58.895 ","End":"03:03.990","Text":"which is square root of x plus 1 plus 2."},{"Start":"03:03.990 ","End":"03:06.105","Text":"But I can\u0027t just multiply it,"},{"Start":"03:06.105 ","End":"03:09.680","Text":"I have to compensate by multiplying the numerator also."},{"Start":"03:09.680 ","End":"03:14.375","Text":"Here again, I\u0027m going to write the square root of x plus 1 plus 2."},{"Start":"03:14.375 ","End":"03:18.650","Text":"In fact, this whole thing is just equal to 1."},{"Start":"03:18.650 ","End":"03:22.235","Text":"So I can multiply, haven\u0027t changed anything."},{"Start":"03:22.235 ","End":"03:29.450","Text":"Now, this equals the limit as x tends to 3."},{"Start":"03:29.450 ","End":"03:32.420","Text":"I\u0027ll do the denominator first"},{"Start":"03:32.420 ","End":"03:33.995","Text":"because we\u0027ve actually done that."},{"Start":"03:33.995 ","End":"03:36.245","Text":"This times this, we\u0027ve already done up here,"},{"Start":"03:36.245 ","End":"03:38.195","Text":"and we get x minus 3."},{"Start":"03:38.195 ","End":"03:40.895","Text":"The numerator, this bit times this bit,"},{"Start":"03:40.895 ","End":"03:46.505","Text":"I\u0027ve got square root of x plus 1 plus 2,"},{"Start":"03:46.505 ","End":"03:50.285","Text":"and look we have x minus 3 here and x minus 3 here."},{"Start":"03:50.285 ","End":"03:53.120","Text":"We can cancel this with this."},{"Start":"03:53.120 ","End":"03:55.819","Text":"We haven\u0027t canceled by 0,"},{"Start":"03:55.819 ","End":"03:59.180","Text":"because the x tends to 3 but doesn\u0027t equal 3,"},{"Start":"03:59.180 ","End":"04:01.190","Text":"and that\u0027s why this is not 0."},{"Start":"04:01.190 ","End":"04:04.610","Text":"All we\u0027re left with is this bit in the brackets,"},{"Start":"04:04.610 ","End":"04:06.980","Text":"square root of x plus 1 plus 2."},{"Start":"04:06.980 ","End":"04:09.660","Text":"That\u0027s certainly elementary function,"},{"Start":"04:09.660 ","End":"04:12.345","Text":"and this time we can substitute 3."},{"Start":"04:12.345 ","End":"04:17.750","Text":"This is going to equal just substituting 3 for x,"},{"Start":"04:17.750 ","End":"04:24.260","Text":"we get the square root of 3 plus 1 plus 2,"},{"Start":"04:24.260 ","End":"04:26.435","Text":"which is 4,"},{"Start":"04:26.435 ","End":"04:29.550","Text":"and that\u0027s the answer."}],"ID":1545},{"Watched":false,"Name":"Exercise 3","Duration":"5m 22s","ChapterTopicVideoID":1534,"CourseChapterTopicPlaylistID":164,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.705","Text":"In this exercise, we want to find the limit as x tends to 3 of this function,"},{"Start":"00:06.705 ","End":"00:12.120","Text":"3 minus the square root of x plus 6 over 2x minus 6."},{"Start":"00:12.120 ","End":"00:15.000","Text":"It\u0027s elementary as a function."},{"Start":"00:15.000 ","End":"00:19.425","Text":"Hopefully, we could just substitute x equals 3."},{"Start":"00:19.425 ","End":"00:25.410","Text":"Unfortunately, that doesn\u0027t work because if we put x equals 3 in the denominator,"},{"Start":"00:25.410 ","End":"00:26.460","Text":"we get 0,"},{"Start":"00:26.460 ","End":"00:28.935","Text":"twice 3 minus 6 is 0."},{"Start":"00:28.935 ","End":"00:31.590","Text":"In the numerator,"},{"Start":"00:31.590 ","End":"00:34.935","Text":"we\u0027re in a 0 over 0 situation."},{"Start":"00:34.935 ","End":"00:40.970","Text":"We have to see what tools we could use to help us do this, simplify it."},{"Start":"00:40.970 ","End":"00:45.665","Text":"I notice that there\u0027s a square root in 1 of the terms in the numerator."},{"Start":"00:45.665 ","End":"00:48.485","Text":"Whenever there\u0027s a square root or more"},{"Start":"00:48.485 ","End":"00:51.590","Text":"in one of the terms in the numerator or denominator,"},{"Start":"00:51.590 ","End":"00:55.505","Text":"it usually indicates that we should try the method of conjugates."},{"Start":"00:55.505 ","End":"00:58.250","Text":"I\u0027ll remind you what a conjugate is,"},{"Start":"00:58.250 ","End":"01:01.220","Text":"in general, of a sum or difference, let\u0027s say,"},{"Start":"01:01.220 ","End":"01:05.000","Text":"A plus B is just the same thing with the opposite sign,"},{"Start":"01:05.000 ","End":"01:07.085","Text":"if it\u0027s a plus, it\u0027s a minus,"},{"Start":"01:07.085 ","End":"01:09.925","Text":"is A minus B."},{"Start":"01:09.925 ","End":"01:12.840","Text":"In other words, each is a conjugate of the other."},{"Start":"01:12.840 ","End":"01:16.610","Text":"The thing about conjugates that makes them useful is that"},{"Start":"01:16.610 ","End":"01:24.370","Text":"if you multiply a conjugate pair, A plus B times A minus B,"},{"Start":"01:24.370 ","End":"01:30.905","Text":"we get, using the difference of squares formula, A squared minus B squared."},{"Start":"01:30.905 ","End":"01:33.770","Text":"If either A or B is a square root,"},{"Start":"01:33.770 ","End":"01:36.080","Text":"when we square it, we get rid of the square root,"},{"Start":"01:36.080 ","End":"01:38.220","Text":"so this could be useful."},{"Start":"01:38.220 ","End":"01:41.570","Text":"Let\u0027s see what happens in our case."},{"Start":"01:41.570 ","End":"01:44.375","Text":"The numerator is the one with the square root."},{"Start":"01:44.375 ","End":"01:49.445","Text":"What we\u0027d like to do is to multiply the numerator by its conjugate."},{"Start":"01:49.445 ","End":"01:53.760","Text":"Let\u0027s just do that as a side exercise first."},{"Start":"01:53.760 ","End":"01:55.320","Text":"This is A minus B,"},{"Start":"01:55.320 ","End":"01:57.090","Text":"we multiply by A plus B,"},{"Start":"01:57.090 ","End":"02:06.185","Text":"we get 3 minus the square root of x plus 6 times its conjugate,"},{"Start":"02:06.185 ","End":"02:09.425","Text":"which is if it\u0027s a minus, it\u0027s a plus, and vice versa,"},{"Start":"02:09.425 ","End":"02:13.845","Text":"3 plus the square root of x plus 6."},{"Start":"02:13.845 ","End":"02:14.940","Text":"This is equal to,"},{"Start":"02:14.940 ","End":"02:18.065","Text":"using this formula of A squared minus B squared,"},{"Start":"02:18.065 ","End":"02:25.740","Text":"A squared is 3 squared is 9 minus, the other part squared,"},{"Start":"02:25.740 ","End":"02:30.860","Text":"so the square root comes off and we\u0027re just left with, x plus 6."},{"Start":"02:30.860 ","End":"02:36.280","Text":"This simplifies to 3 minus x."},{"Start":"02:36.280 ","End":"02:41.030","Text":"Now, let\u0027s try using this to help us to solve this limit,"},{"Start":"02:41.030 ","End":"02:42.860","Text":"so I\u0027ll rewrite it."},{"Start":"02:42.860 ","End":"02:45.350","Text":"What I really want to do is to multiply"},{"Start":"02:45.350 ","End":"02:49.700","Text":"this 3 minus the square root by 3 plus the square root by its conjugate."},{"Start":"02:49.700 ","End":"02:55.950","Text":"I\u0027m going to multiply by 3 plus the square root of x plus 6."},{"Start":"02:55.950 ","End":"03:00.765","Text":"But I can\u0027t just multiply because I\u0027ve changed the exercise."},{"Start":"03:00.765 ","End":"03:06.910","Text":"But if I multiply the denominator also by the same thing,"},{"Start":"03:06.910 ","End":"03:11.610","Text":"this whole thing is just equal to 1,"},{"Start":"03:11.610 ","End":"03:18.435","Text":"and I can always multiply by 1 without changing anything, so that\u0027s allowed."},{"Start":"03:18.435 ","End":"03:23.290","Text":"What we get is limit x tends to 3."},{"Start":"03:23.290 ","End":"03:27.300","Text":"Start with the numerator because we\u0027ve already done this."},{"Start":"03:27.300 ","End":"03:31.760","Text":"This numerator times this numerator is exactly what we did here,"},{"Start":"03:31.760 ","End":"03:35.509","Text":"so we can write the answer as 3 minus x."},{"Start":"03:35.509 ","End":"03:38.060","Text":"Then in the denominator,"},{"Start":"03:38.060 ","End":"03:41.410","Text":"we just have to multiply this by this,"},{"Start":"03:41.410 ","End":"03:47.235","Text":"2x minus 6 times"},{"Start":"03:47.235 ","End":"03:53.595","Text":"3 plus the square root of x plus 6."},{"Start":"03:53.595 ","End":"03:56.100","Text":"Now, previously in exercises of this sort,"},{"Start":"03:56.100 ","End":"03:58.005","Text":"there was a factor that canceled,"},{"Start":"03:58.005 ","End":"04:03.170","Text":"and here, it doesn\u0027t seem to be a factor that appears both in numerator and denominator."},{"Start":"04:03.170 ","End":"04:05.000","Text":"But if you look more closely,"},{"Start":"04:05.000 ","End":"04:08.525","Text":"2x minus 6 is not very different from 3 minus x."},{"Start":"04:08.525 ","End":"04:10.235","Text":"If we reverse the order,"},{"Start":"04:10.235 ","End":"04:12.190","Text":"it was x minus 3,"},{"Start":"04:12.190 ","End":"04:14.265","Text":"this is exactly double this."},{"Start":"04:14.265 ","End":"04:20.810","Text":"I\u0027d like to make a note at the side that 2x minus 6 is"},{"Start":"04:20.810 ","End":"04:28.620","Text":"simply equal to minus twice 3 minus x."},{"Start":"04:28.620 ","End":"04:30.815","Text":"If I notice this,"},{"Start":"04:30.815 ","End":"04:33.855","Text":"then I can do a cancellation of sorts."},{"Start":"04:33.855 ","End":"04:37.759","Text":"I can cancel this with this,"},{"Start":"04:37.759 ","End":"04:42.295","Text":"but I have to leave a minus 2 here."},{"Start":"04:42.295 ","End":"04:48.090","Text":"Continuing, this a 1 in the numerator,"},{"Start":"04:48.090 ","End":"04:52.855","Text":"all I have is 1 over minus twice 3 plus the square root."},{"Start":"04:52.855 ","End":"04:55.190","Text":"This is also an elementary function,"},{"Start":"04:55.190 ","End":"04:58.850","Text":"but this time we can substitute x equals 3."},{"Start":"04:58.850 ","End":"05:00.455","Text":"If we do that,"},{"Start":"05:00.455 ","End":"05:05.765","Text":"we\u0027ll get the answer which is 1 over"},{"Start":"05:05.765 ","End":"05:15.035","Text":"minus 2 times 3 plus square root of 3 plus 6,"},{"Start":"05:15.035 ","End":"05:20.790","Text":"so the answer is just minus 1 over 12."},{"Start":"05:20.790 ","End":"05:23.890","Text":"That\u0027s the answer. We\u0027re done."}],"ID":1546},{"Watched":false,"Name":"Exercise 4","Duration":"8m 28s","ChapterTopicVideoID":1535,"CourseChapterTopicPlaylistID":164,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.135","Text":"In this exercise, we\u0027re asked to find the limit as x tends to 1 of the function,"},{"Start":"00:06.135 ","End":"00:10.515","Text":"square root of x squared plus x plus 2 minus 2,"},{"Start":"00:10.515 ","End":"00:13.095","Text":"all over x squared minus 1."},{"Start":"00:13.095 ","End":"00:15.255","Text":"This is an elementary function,"},{"Start":"00:15.255 ","End":"00:19.830","Text":"we would like to hope that we could just substitute x equals 1."},{"Start":"00:19.830 ","End":"00:24.285","Text":"Unfortunately, that doesn\u0027t work because if we do substitute x equals 1,"},{"Start":"00:24.285 ","End":"00:26.400","Text":"we get 1 squared minus 1,"},{"Start":"00:26.400 ","End":"00:30.915","Text":"which is 0, and in the numerator is 0."},{"Start":"00:30.915 ","End":"00:33.660","Text":"We\u0027re in the 0 over 0 situation"},{"Start":"00:33.660 ","End":"00:36.195","Text":"and we have to see what tools we can use."},{"Start":"00:36.195 ","End":"00:39.480","Text":"Well, the biggest hint is that there is a square root here,"},{"Start":"00:39.480 ","End":"00:41.960","Text":"and there\u0027s a couple of terms in the numerator,"},{"Start":"00:41.960 ","End":"00:43.910","Text":"a couple of terms in the denominator,"},{"Start":"00:43.910 ","End":"00:46.070","Text":"and the square root."},{"Start":"00:46.070 ","End":"00:49.810","Text":"This usually indicates the use of the conjugate,"},{"Start":"00:49.810 ","End":"00:51.200","Text":"and in case you forgotten,"},{"Start":"00:51.200 ","End":"00:53.465","Text":"I\u0027ll remind you what a conjugate is."},{"Start":"00:53.465 ","End":"00:58.895","Text":"In general, if you have a term of the form A plus B,"},{"Start":"00:58.895 ","End":"01:01.205","Text":"then I\u0027ll put it in brackets,"},{"Start":"01:01.205 ","End":"01:03.635","Text":"it has a conjugate term,"},{"Start":"01:03.635 ","End":"01:06.080","Text":"which is A minus B."},{"Start":"01:06.080 ","End":"01:09.650","Text":"Just the reverse sign works both ways."},{"Start":"01:09.650 ","End":"01:12.695","Text":"These 2 are conjugates of each other."},{"Start":"01:12.695 ","End":"01:16.930","Text":"This is useful because if you multiply 2 conjugates,"},{"Start":"01:16.930 ","End":"01:21.960","Text":"in this case A plus B by A minus B,"},{"Start":"01:21.960 ","End":"01:27.230","Text":"you get A squared minus B squared using the difference of squares formula,"},{"Start":"01:27.230 ","End":"01:31.085","Text":"and if A or B happens to be a square root,"},{"Start":"01:31.085 ","End":"01:33.830","Text":"which it will be, then when you square it,"},{"Start":"01:33.830 ","End":"01:35.975","Text":"the square root will just drop off."},{"Start":"01:35.975 ","End":"01:37.370","Text":"That will be helpful."},{"Start":"01:37.370 ","End":"01:40.285","Text":"Let\u0027s see how it helps us in our case."},{"Start":"01:40.285 ","End":"01:42.705","Text":"The numerator is the 1 with the square root,"},{"Start":"01:42.705 ","End":"01:45.950","Text":"so let\u0027s try and see what happens if we"},{"Start":"01:45.950 ","End":"01:49.310","Text":"look at the conjugate of the numerator and then multiply."},{"Start":"01:49.310 ","End":"01:52.160","Text":"We\u0027ll do this as a side exercise at first,"},{"Start":"01:52.160 ","End":"02:00.600","Text":"so take the numerator of x squared plus x plus 2 minus 2."},{"Start":"02:00.600 ","End":"02:03.375","Text":"That\u0027s like the A minus B,"},{"Start":"02:03.375 ","End":"02:07.490","Text":"and its conjugate will be the same thing with a plus,"},{"Start":"02:07.490 ","End":"02:15.754","Text":"so square root of x squared plus x plus 2, this time plus 2."},{"Start":"02:15.754 ","End":"02:18.215","Text":"If we multiply this out,"},{"Start":"02:18.215 ","End":"02:20.550","Text":"we get the A squared,"},{"Start":"02:20.550 ","End":"02:21.770","Text":"which is this 1 squared,"},{"Start":"02:21.770 ","End":"02:25.265","Text":"which is x squared plus x plus 2,"},{"Start":"02:25.265 ","End":"02:28.700","Text":"because a square root drops off, minus 2 squared."},{"Start":"02:28.700 ","End":"02:30.095","Text":"I\u0027ll just write it like that."},{"Start":"02:30.095 ","End":"02:31.340","Text":"That\u0027s obviously 4,"},{"Start":"02:31.340 ","End":"02:37.320","Text":"but we end up getting x squared plus x minus 2,"},{"Start":"02:37.320 ","End":"02:40.720","Text":"because it is 2 minus 4 is minus 2."},{"Start":"02:40.910 ","End":"02:44.030","Text":"That\u0027s a side exercise."},{"Start":"02:44.030 ","End":"02:46.445","Text":"Now see how it helps us solve our limit."},{"Start":"02:46.445 ","End":"02:48.215","Text":"Let\u0027s look again the limit."},{"Start":"02:48.215 ","End":"02:54.270","Text":"The limit as x tends to 1 of,"},{"Start":"02:54.270 ","End":"02:55.880","Text":"just copy it out again."},{"Start":"02:55.880 ","End":"02:57.289","Text":"To make things simple,"},{"Start":"02:57.289 ","End":"03:02.825","Text":"what I want to do is to multiply this thing by the conjugate,"},{"Start":"03:02.825 ","End":"03:04.130","Text":"like we did above,"},{"Start":"03:04.130 ","End":"03:10.300","Text":"which is the square root of x squared plus x plus 2 plus 2."},{"Start":"03:10.300 ","End":"03:13.740","Text":"But I can\u0027t just multiply the numerator,"},{"Start":"03:13.740 ","End":"03:15.500","Text":"that changes the exercise."},{"Start":"03:15.500 ","End":"03:21.050","Text":"However, if I also multiply the denominator by the same thing,"},{"Start":"03:21.050 ","End":"03:23.225","Text":"just copy it out again."},{"Start":"03:23.225 ","End":"03:28.235","Text":"Then essentially I multiply this fraction by this whole thing,"},{"Start":"03:28.235 ","End":"03:30.335","Text":"which happens to equal 1,"},{"Start":"03:30.335 ","End":"03:32.540","Text":"so I haven\u0027t changed anything."},{"Start":"03:32.540 ","End":"03:35.350","Text":"Let\u0027s continue and see what this is equal to."},{"Start":"03:35.350 ","End":"03:39.915","Text":"This is equal to the limit as x goes to 1."},{"Start":"03:39.915 ","End":"03:41.690","Text":"This times this we\u0027ve already done,"},{"Start":"03:41.690 ","End":"03:43.040","Text":"that\u0027s what we just did here,"},{"Start":"03:43.040 ","End":"03:50.600","Text":"so we can write the final answer for the numerator is x squared plus x minus 2,"},{"Start":"03:50.600 ","End":"03:52.125","Text":"and on the denominator,"},{"Start":"03:52.125 ","End":"03:54.165","Text":"something a bit more complicated,"},{"Start":"03:54.165 ","End":"04:00.664","Text":"x squared minus 1 times, in brackets,"},{"Start":"04:00.664 ","End":"04:09.720","Text":"the square root of x squared plus x plus 2 plus 2."},{"Start":"04:09.720 ","End":"04:12.875","Text":"Now, hopefully, we\u0027d like to cancel something,"},{"Start":"04:12.875 ","End":"04:16.665","Text":"but there\u0027s no obvious factor to cancel,"},{"Start":"04:16.665 ","End":"04:21.290","Text":"and usually, the idea is to factor quadratic expressions,"},{"Start":"04:21.290 ","End":"04:22.580","Text":"or whatever we can factor,"},{"Start":"04:22.580 ","End":"04:24.980","Text":"and hopefully, these things will have a common factor."},{"Start":"04:24.980 ","End":"04:28.590","Text":"Now, we know how to factor quadratic expressions."},{"Start":"04:28.590 ","End":"04:29.750","Text":"Here we have a quadratic,"},{"Start":"04:29.750 ","End":"04:31.205","Text":"here we have a quadratic."},{"Start":"04:31.205 ","End":"04:37.310","Text":"Let\u0027s take time off to just factorize these and then return to the main exercise."},{"Start":"04:37.310 ","End":"04:40.055","Text":"I want to remind you of another formula,"},{"Start":"04:40.055 ","End":"04:44.030","Text":"and I\u0027ll write it at the side, and that is,"},{"Start":"04:44.030 ","End":"04:51.575","Text":"that if we have an expression such as x squared plus bx plus c,"},{"Start":"04:51.575 ","End":"04:56.710","Text":"then this factors to x minus"},{"Start":"04:56.710 ","End":"05:02.220","Text":"x_1 times x minus x_2,"},{"Start":"05:02.220 ","End":"05:10.785","Text":"where x_1 and x_2 are the roots of the quadratic equation."},{"Start":"05:10.785 ","End":"05:14.150","Text":"Let\u0027s write it the roots of the equation by which"},{"Start":"05:14.150 ","End":"05:17.660","Text":"I mean x squared plus bx plus c equals 0."},{"Start":"05:17.660 ","End":"05:19.085","Text":"There as we solved this,"},{"Start":"05:19.085 ","End":"05:21.890","Text":"get the 2 roots, and then we can factor it this way."},{"Start":"05:21.890 ","End":"05:24.575","Text":"Let\u0027s see what happens in our case."},{"Start":"05:24.575 ","End":"05:26.395","Text":"We\u0027ll use this twice,"},{"Start":"05:26.395 ","End":"05:32.495","Text":"once on the numerator and also on the denominator."},{"Start":"05:32.495 ","End":"05:35.675","Text":"Let\u0027s do the green 1 first."},{"Start":"05:35.675 ","End":"05:38.870","Text":"First of all, we solve the equation,"},{"Start":"05:38.870 ","End":"05:43.715","Text":"x squared minus 1 equals 0."},{"Start":"05:43.715 ","End":"05:44.900","Text":"Well, in this case,"},{"Start":"05:44.900 ","End":"05:47.390","Text":"we didn\u0027t really need to do it this way,"},{"Start":"05:47.390 ","End":"05:49.280","Text":"there\u0027s actually a shorter way."},{"Start":"05:49.280 ","End":"05:53.210","Text":"We could have solved it and got that x is plus or minus 1,"},{"Start":"05:53.210 ","End":"05:55.325","Text":"but the easiest thing maybe,"},{"Start":"05:55.325 ","End":"05:58.160","Text":"was just to notice that this is also a difference of squares,"},{"Start":"05:58.160 ","End":"06:00.475","Text":"x squared minus 1 squared."},{"Start":"06:00.475 ","End":"06:09.140","Text":"X squared minus 1 is going to equal x minus 1 times x plus 1."},{"Start":"06:09.140 ","End":"06:14.320","Text":"The more slightly trickier 1 is the blue 1."},{"Start":"06:14.320 ","End":"06:22.300","Text":"To do that, we have to solve x squared plus x minus 2 equals 0."},{"Start":"06:22.300 ","End":"06:25.465","Text":"I assume you know how to solve quadratic equations."},{"Start":"06:25.465 ","End":"06:26.860","Text":"I won\u0027t solve it for you,"},{"Start":"06:26.860 ","End":"06:28.930","Text":"I\u0027ll just tell you that the roots are,"},{"Start":"06:28.930 ","End":"06:30.835","Text":"those are the 2 solutions."},{"Start":"06:30.835 ","End":"06:36.940","Text":"Because of this, it means that just as we go from here to here,"},{"Start":"06:36.940 ","End":"06:45.305","Text":"this means that x squared plus x minus 2 is equal to x minus x_1 minus x_2,"},{"Start":"06:45.305 ","End":"06:47.755","Text":"so it\u0027s x minus 1."},{"Start":"06:47.755 ","End":"06:53.445","Text":"X minus minus 2 makes it x plus 2."},{"Start":"06:53.445 ","End":"06:56.030","Text":"We\u0027ve got for the denominator,"},{"Start":"06:56.030 ","End":"06:59.360","Text":"this factorization, and for the numerator,"},{"Start":"06:59.360 ","End":"07:02.180","Text":"we\u0027re going to use this factorization,"},{"Start":"07:02.180 ","End":"07:07.580","Text":"and let\u0027s go back to writing what the limit equals."},{"Start":"07:07.580 ","End":"07:10.355","Text":"I\u0027ll just put an arrow here,"},{"Start":"07:10.355 ","End":"07:16.100","Text":"and this is going to equal the limit x goes to 1."},{"Start":"07:16.100 ","End":"07:18.635","Text":"Here, I\u0027ve factorized it,"},{"Start":"07:18.635 ","End":"07:20.750","Text":"the blue 1, x minus 1,"},{"Start":"07:20.750 ","End":"07:23.165","Text":"x plus 2 over."},{"Start":"07:23.165 ","End":"07:25.070","Text":"Now the green 1,"},{"Start":"07:25.070 ","End":"07:27.515","Text":"x minus 1, x plus 1,"},{"Start":"07:27.515 ","End":"07:29.240","Text":"and this bit here,"},{"Start":"07:29.240 ","End":"07:32.015","Text":"which we have to also maintain,"},{"Start":"07:32.015 ","End":"07:35.134","Text":"now we can definitely see something to cancel."},{"Start":"07:35.134 ","End":"07:39.195","Text":"The x minus 1 goes with the x minus 1."},{"Start":"07:39.195 ","End":"07:44.180","Text":"Notice that x minus 1 is not 0 because x is not equal to 1,"},{"Start":"07:44.180 ","End":"07:45.500","Text":"it tends to 1"},{"Start":"07:45.500 ","End":"07:46.985","Text":"but it\u0027s not equal to 1."},{"Start":"07:46.985 ","End":"07:51.815","Text":"What we\u0027re left with is the limit as x goes to 1 of this function here,"},{"Start":"07:51.815 ","End":"07:56.210","Text":"and here there is no problem in substituting x equals 1,"},{"Start":"07:56.210 ","End":"08:01.835","Text":"so this time it is equal to just substituting x equals 1,"},{"Start":"08:01.835 ","End":"08:04.000","Text":"so let\u0027s see, I\u0027ll just write it underneath here."},{"Start":"08:04.000 ","End":"08:10.560","Text":"We get 1 plus 2 over 1 plus 1 times"},{"Start":"08:10.560 ","End":"08:18.420","Text":"the square root of 1 plus 1 plus 2 plus 2,"},{"Start":"08:18.420 ","End":"08:24.150","Text":"and let\u0027s see what this gives us. 1 plus 2 is 3,"},{"Start":"08:24.150 ","End":"08:29.920","Text":"so the answer is 3/8, and we\u0027re done."}],"ID":1547},{"Watched":false,"Name":"Exercise 5","Duration":"6m 9s","ChapterTopicVideoID":1536,"CourseChapterTopicPlaylistID":164,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.245","Text":"In this exercise, we have to find the limit as x tends to 4 of the following function,"},{"Start":"00:07.245 ","End":"00:10.935","Text":"square root of 2x plus 1 minus the square root of x plus 5,"},{"Start":"00:10.935 ","End":"00:13.320","Text":"all over x minus 4."},{"Start":"00:13.320 ","End":"00:17.145","Text":"If we were able to substitute x equals 4 here,"},{"Start":"00:17.145 ","End":"00:19.860","Text":"that would be very nice and we\u0027d be done."},{"Start":"00:19.860 ","End":"00:22.755","Text":"But unfortunately, this is not the case."},{"Start":"00:22.755 ","End":"00:26.505","Text":"If we put x equals 4 in the denominator,"},{"Start":"00:26.505 ","End":"00:30.840","Text":"we get 4 minus 4, which is 0."},{"Start":"00:30.840 ","End":"00:33.870","Text":"If we try putting x equals 4 in the numerator, let\u0027s see,"},{"Start":"00:33.870 ","End":"00:36.435","Text":"twice 4 plus 1 is 9,"},{"Start":"00:36.435 ","End":"00:39.600","Text":"so we have square root of 9,"},{"Start":"00:39.600 ","End":"00:42.460","Text":"and 4 plus 5 is 9."},{"Start":"00:42.460 ","End":"00:46.595","Text":"We have also square root of 9 and it\u0027s a minus,"},{"Start":"00:46.595 ","End":"00:48.850","Text":"so it also gives us 0."},{"Start":"00:48.850 ","End":"00:52.680","Text":"Basically, we\u0027re in a 0 over 0 situation,"},{"Start":"00:52.680 ","End":"00:56.930","Text":"so we have to think of what techniques we know."},{"Start":"00:56.930 ","End":"00:59.330","Text":"The biggest hint is the square roots."},{"Start":"00:59.330 ","End":"01:03.365","Text":"When you have a term and 1 or more of them has a square root,"},{"Start":"01:03.365 ","End":"01:07.920","Text":"usually, it indicates the use of the conjugate."},{"Start":"01:07.930 ","End":"01:11.795","Text":"I want to remind you what a conjugate is."},{"Start":"01:11.795 ","End":"01:13.490","Text":"When you have a term,"},{"Start":"01:13.490 ","End":"01:19.250","Text":"an expression of the form A minus B or A plus B,"},{"Start":"01:19.250 ","End":"01:27.050","Text":"its conjugate is the 1 with the opposite sign and the conjugate of A minus B is A plus B."},{"Start":"01:27.050 ","End":"01:29.230","Text":"Each of these is the conjugate of the other,"},{"Start":"01:29.230 ","End":"01:32.840","Text":"so these are conjugates."},{"Start":"01:32.840 ","End":"01:35.225","Text":"Now, the useful thing about conjugates,"},{"Start":"01:35.225 ","End":"01:37.490","Text":"that if you multiply a pair of them,"},{"Start":"01:37.490 ","End":"01:39.460","Text":"let\u0027s see what we get."},{"Start":"01:39.460 ","End":"01:43.040","Text":"A minus B times"},{"Start":"01:43.040 ","End":"01:50.315","Text":"A plus B is equal to A squared minus B squared."},{"Start":"01:50.315 ","End":"01:54.230","Text":"There\u0027s a difference of squares formula which gives us this."},{"Start":"01:54.230 ","End":"01:58.595","Text":"Now, if A or B or both were square roots,"},{"Start":"01:58.595 ","End":"01:59.960","Text":"when we square them,"},{"Start":"01:59.960 ","End":"02:01.240","Text":"we get rid of the square root,"},{"Start":"02:01.240 ","End":"02:03.710","Text":"so that\u0027s why conjugates are useful."},{"Start":"02:03.710 ","End":"02:06.470","Text":"Let\u0027s see how they can help us in our case."},{"Start":"02:06.470 ","End":"02:11.720","Text":"Here, the numerator is the 1 that has square roots, so its numerator,"},{"Start":"02:11.720 ","End":"02:14.389","Text":"that\u0027s interesting and let\u0027s just as a side exercise,"},{"Start":"02:14.389 ","End":"02:19.520","Text":"see what would happen if we multiply this numerator here by its conjugate."},{"Start":"02:19.520 ","End":"02:21.260","Text":"In this case, the A minus B,"},{"Start":"02:21.260 ","End":"02:24.615","Text":"this is A minus B will multiply it by A plus B."},{"Start":"02:24.615 ","End":"02:25.785","Text":"Let\u0027s see what happens."},{"Start":"02:25.785 ","End":"02:34.805","Text":"Square root of 2x plus 1 minus the square root of x plus 5."},{"Start":"02:34.805 ","End":"02:36.020","Text":"I\u0027m going to multiply,"},{"Start":"02:36.020 ","End":"02:37.610","Text":"so I\u0027ll need brackets,"},{"Start":"02:37.610 ","End":"02:40.465","Text":"and for the other, for the conjugate."},{"Start":"02:40.465 ","End":"02:42.775","Text":"Square root, same thing,"},{"Start":"02:42.775 ","End":"02:45.000","Text":"2x plus 1,"},{"Start":"02:45.000 ","End":"02:47.310","Text":"but this time with a plus in the middle,"},{"Start":"02:47.310 ","End":"02:52.130","Text":"and again, square root of x plus 5."},{"Start":"02:52.130 ","End":"02:54.260","Text":"Let\u0027s see what this gives us."},{"Start":"02:54.260 ","End":"02:56.300","Text":"If we look at this, A minus B,"},{"Start":"02:56.300 ","End":"02:58.690","Text":"A plus B is A squared minus B squared,"},{"Start":"02:58.690 ","End":"03:01.925","Text":"so our A squared is just the square root squared,"},{"Start":"03:01.925 ","End":"03:06.190","Text":"so this is just 2x plus 1,"},{"Start":"03:06.190 ","End":"03:08.040","Text":"which was the first,"},{"Start":"03:08.040 ","End":"03:10.415","Text":"minus same thing here."},{"Start":"03:10.415 ","End":"03:16.235","Text":"Just take up the square root minus x plus 5 with no square root,"},{"Start":"03:16.235 ","End":"03:25.110","Text":"and we just collect terms 2x minus x is x and 1 minus 5 is minus 4."},{"Start":"03:25.110 ","End":"03:31.439","Text":"This is just equal to x minus 4."},{"Start":"03:31.439 ","End":"03:35.765","Text":"Now we want to see how this is going to help us in finding our limit,"},{"Start":"03:35.765 ","End":"03:39.245","Text":"the limit, let me just write the exercise again,"},{"Start":"03:39.245 ","End":"03:41.450","Text":"x goes to 4."},{"Start":"03:41.450 ","End":"03:45.450","Text":"What we\u0027re going to do is to multiply,"},{"Start":"03:45.450 ","End":"03:48.680","Text":"we want to multiply the numerator by its conjugate,"},{"Start":"03:48.680 ","End":"03:50.300","Text":"which will give us a nice expressions."},{"Start":"03:50.300 ","End":"03:54.830","Text":"We multiply by square root of"},{"Start":"03:54.830 ","End":"04:02.385","Text":"2x plus 1 plus square root of x plus 5."},{"Start":"04:02.385 ","End":"04:05.540","Text":"I can\u0027t go ahead multiplying by whatever I want,"},{"Start":"04:05.540 ","End":"04:07.039","Text":"because I\u0027ve changed the exercise."},{"Start":"04:07.039 ","End":"04:13.140","Text":"But if I multiply top and bottom by the same thing,"},{"Start":"04:13.180 ","End":"04:18.170","Text":"this second fraction as a whole is just equal to 1,"},{"Start":"04:18.170 ","End":"04:21.320","Text":"A over A is 1, whatever A is."},{"Start":"04:21.320 ","End":"04:23.945","Text":"We haven\u0027t changed anything,"},{"Start":"04:23.945 ","End":"04:26.720","Text":"and we can remember to multiply fractions,"},{"Start":"04:26.720 ","End":"04:30.140","Text":"we multiply numerators, and we multiply denominators."},{"Start":"04:30.140 ","End":"04:37.140","Text":"What we get is the limit as x approaches 4 of,"},{"Start":"04:37.140 ","End":"04:40.640","Text":"this times this is an exercise we\u0027ve already done up here."},{"Start":"04:40.640 ","End":"04:45.625","Text":"The whole numerator just is x minus 4,"},{"Start":"04:45.625 ","End":"04:49.790","Text":"and on the denominator we have this x minus 4,"},{"Start":"04:49.790 ","End":"04:51.320","Text":"let me write that in brackets"},{"Start":"04:51.320 ","End":"04:53.815","Text":"because the denominator is like a bracket,"},{"Start":"04:53.815 ","End":"04:57.095","Text":"times whatever is left here,"},{"Start":"04:57.095 ","End":"05:07.550","Text":"which is the square root of 2x plus 1 plus the square root of x plus 5."},{"Start":"05:07.550 ","End":"05:11.720","Text":"Well, we are lucky that something cancels,"},{"Start":"05:11.720 ","End":"05:14.780","Text":"the x minus 4 cancels with the x minus 4,"},{"Start":"05:14.780 ","End":"05:17.845","Text":"leaving just a 1 in the numerator."},{"Start":"05:17.845 ","End":"05:21.905","Text":"Now we have to find the limit of this function,"},{"Start":"05:21.905 ","End":"05:23.810","Text":"which is also elementary,"},{"Start":"05:23.810 ","End":"05:27.790","Text":"but in this case, we can substitute x equals 4,"},{"Start":"05:27.790 ","End":"05:30.235","Text":"and this equals,"},{"Start":"05:30.235 ","End":"05:32.765","Text":"wherever I see x I put 4."},{"Start":"05:32.765 ","End":"05:41.700","Text":"I\u0027m going to get 1 over the square root of 2 times 4 plus 1,"},{"Start":"05:41.890 ","End":"05:49.535","Text":"plus the square root of 4 plus 5."},{"Start":"05:49.535 ","End":"05:51.155","Text":"Let\u0027s see what this equals,"},{"Start":"05:51.155 ","End":"05:53.300","Text":"square root of 9 is 3,"},{"Start":"05:53.300 ","End":"05:55.460","Text":"and this denominator is 3,"},{"Start":"05:55.460 ","End":"05:59.445","Text":"and the square root of 4 plus 5 is also 9,"},{"Start":"05:59.445 ","End":"06:01.545","Text":"square root of 9 is 3."},{"Start":"06:01.545 ","End":"06:04.050","Text":"We have 1 over 3 plus 3,"},{"Start":"06:04.050 ","End":"06:09.430","Text":"1/6, and that\u0027s our answer."}],"ID":1548},{"Watched":false,"Name":"Exercise 6","Duration":"8m 9s","ChapterTopicVideoID":1537,"CourseChapterTopicPlaylistID":164,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this exercise, we have to find the limit"},{"Start":"00:03.030 ","End":"00:09.240","Text":"as x tends to 1 of the function 2 minus square root of 3x plus 1"},{"Start":"00:09.240 ","End":"00:12.990","Text":"over 1 minus the square root of 2x minus 1."},{"Start":"00:12.990 ","End":"00:14.925","Text":"Now, this is an elementary function,"},{"Start":"00:14.925 ","End":"00:16.785","Text":"and in such cases,"},{"Start":"00:16.785 ","End":"00:20.145","Text":"usually we can just substitute x equals 1."},{"Start":"00:20.145 ","End":"00:22.185","Text":"However, in our case,"},{"Start":"00:22.185 ","End":"00:23.565","Text":"if we try doing that,"},{"Start":"00:23.565 ","End":"00:29.445","Text":"we would get twice 1 minus 1 is 1 square root of 1 is 1,"},{"Start":"00:29.445 ","End":"00:31.620","Text":"1 minus 1 is 0."},{"Start":"00:31.620 ","End":"00:36.410","Text":"In the numerator, 3 times 1 plus 1 is 4."},{"Start":"00:36.410 ","End":"00:37.880","Text":"Square root of 4 is 2."},{"Start":"00:37.880 ","End":"00:39.775","Text":"2 minus 2 is 0."},{"Start":"00:39.775 ","End":"00:43.430","Text":"We are in a 0 over 0 situation."},{"Start":"00:43.430 ","End":"00:47.075","Text":"Actually we can\u0027t substitute x equals 1 unfortunately."},{"Start":"00:47.075 ","End":"00:49.060","Text":"We\u0027re going to have to use other techniques."},{"Start":"00:49.060 ","End":"00:50.315","Text":"Now, taking a look at it,"},{"Start":"00:50.315 ","End":"00:51.810","Text":"we see that there are square roots,"},{"Start":"00:51.810 ","End":"00:56.015","Text":"there are terms with pluses and minuses and some have square roots."},{"Start":"00:56.015 ","End":"01:00.695","Text":"That usually indicates that we should use the technique of conjugates."},{"Start":"01:00.695 ","End":"01:03.955","Text":"I\u0027d like to remind you what a conjugate is."},{"Start":"01:03.955 ","End":"01:08.965","Text":"In general, if I have an expression made up of 2 terms,"},{"Start":"01:08.965 ","End":"01:10.690","Text":"like A plus B,"},{"Start":"01:10.690 ","End":"01:15.740","Text":"its conjugate is A minus B and vice versa."},{"Start":"01:15.740 ","End":"01:19.130","Text":"If I have something of the form A minus B,"},{"Start":"01:19.130 ","End":"01:21.350","Text":"its conjugate is A plus B."},{"Start":"01:21.350 ","End":"01:24.050","Text":"These 2 are conjugate to each other."},{"Start":"01:24.050 ","End":"01:30.320","Text":"The useful thing about them in connection with square roots is that if we multiply them"},{"Start":"01:30.320 ","End":"01:38.825","Text":"A plus B times A minus B is 1 of those formerly called a difference of squares,"},{"Start":"01:38.825 ","End":"01:42.815","Text":"where this is equal to A squared minus B squared."},{"Start":"01:42.815 ","End":"01:48.470","Text":"If A or B or both possibly have square roots after the square,"},{"Start":"01:48.470 ","End":"01:50.420","Text":"they no longer have the square root."},{"Start":"01:50.420 ","End":"01:53.345","Text":"Let\u0027s see how this helps us in our case."},{"Start":"01:53.345 ","End":"01:55.805","Text":"Both in numerator and in denominator,"},{"Start":"01:55.805 ","End":"01:57.365","Text":"we have square roots."},{"Start":"01:57.365 ","End":"02:00.770","Text":"It might be useful to do conjugates twice,"},{"Start":"02:00.770 ","End":"02:03.365","Text":"once for the numerator and once for the denominator."},{"Start":"02:03.365 ","End":"02:06.230","Text":"Let\u0027s do this as a pair of side exercises."},{"Start":"02:06.230 ","End":"02:09.890","Text":"We\u0027ll multiply this by its conjugate and then this by its conjugate,"},{"Start":"02:09.890 ","End":"02:11.825","Text":"and later we\u0027ll tie it all in."},{"Start":"02:11.825 ","End":"02:17.720","Text":"Let\u0027s take first of all the 2 minus the square root of"},{"Start":"02:17.720 ","End":"02:23.450","Text":"3x plus 1 from the numerator and multiply it by its conjugate,"},{"Start":"02:23.450 ","End":"02:24.695","Text":"it\u0027s a minus here,"},{"Start":"02:24.695 ","End":"02:31.325","Text":"so we now need a plus square root of 3x plus 1."},{"Start":"02:31.325 ","End":"02:34.265","Text":"This equals according to this formula,"},{"Start":"02:34.265 ","End":"02:37.505","Text":"first 1 squared minus the second 1 squared."},{"Start":"02:37.505 ","End":"02:39.204","Text":"It\u0027s the 2 squared,"},{"Start":"02:39.204 ","End":"02:42.740","Text":"which is 4 minus the second 1,"},{"Start":"02:42.740 ","End":"02:49.510","Text":"which is now without the square roots, 3x plus 1."},{"Start":"02:49.520 ","End":"02:52.445","Text":"If we simplify this,"},{"Start":"02:52.445 ","End":"02:56.420","Text":"we get 3 minus 3x."},{"Start":"02:56.420 ","End":"03:00.950","Text":"That\u0027s what happens when we multiply the numerator by its conjugate."},{"Start":"03:00.950 ","End":"03:04.265","Text":"Now let\u0027s do the same thing on the denominator which is here."},{"Start":"03:04.265 ","End":"03:10.720","Text":"We\u0027ll take the denominator 1 minus the square root of 2x minus 1,"},{"Start":"03:10.720 ","End":"03:13.005","Text":"and multiply it by its conjugate."},{"Start":"03:13.005 ","End":"03:17.870","Text":"We turn the minus into a plus and we get a similar way."},{"Start":"03:17.870 ","End":"03:20.585","Text":"The first 1 squared minus second 1 squared."},{"Start":"03:20.585 ","End":"03:25.340","Text":"1 squared is 1 less 2x minus"},{"Start":"03:25.340 ","End":"03:31.900","Text":"1 and what we get is 2 minus 2x."},{"Start":"03:33.050 ","End":"03:37.215","Text":"These are side exercises meanwhile."},{"Start":"03:37.215 ","End":"03:39.950","Text":"Let\u0027s see how we can tie them in here."},{"Start":"03:39.950 ","End":"03:44.795","Text":"I\u0027ll continue just by copying the exercise as it was over here."},{"Start":"03:44.795 ","End":"03:47.060","Text":"Now what I\u0027d like to do,"},{"Start":"03:47.060 ","End":"03:50.300","Text":"I\u0027m going to spoil it and then fix it again."},{"Start":"03:50.300 ","End":"03:52.820","Text":"What I\u0027d like to do is multiply this thing,"},{"Start":"03:52.820 ","End":"03:55.385","Text":"the numerator by its conjugate."},{"Start":"03:55.385 ","End":"04:00.540","Text":"Because I know that will give me something nicer without square roots."},{"Start":"04:00.540 ","End":"04:06.180","Text":"2 plus a square root of 3x plus 1."},{"Start":"04:06.180 ","End":"04:12.110","Text":"What I\u0027d like to do also is multiply the denominator by its conjugate,"},{"Start":"04:12.110 ","End":"04:13.790","Text":"like in the second line,"},{"Start":"04:13.790 ","End":"04:20.630","Text":"and that\u0027s 1 plus the square root of 2x minus 1."},{"Start":"04:20.630 ","End":"04:22.340","Text":"That\u0027s what I\u0027d like to do."},{"Start":"04:22.340 ","End":"04:25.430","Text":"But if I multiply this thing by something and then I\u0027ve ruined it,"},{"Start":"04:25.430 ","End":"04:27.560","Text":"so I have to now fix it."},{"Start":"04:27.560 ","End":"04:30.950","Text":"Let me just show you as a side exercise."},{"Start":"04:30.950 ","End":"04:40.025","Text":"If I take a fraction a over b and then multiply it by b over a,"},{"Start":"04:40.025 ","End":"04:42.560","Text":"the b cancels with b, a with a,"},{"Start":"04:42.560 ","End":"04:45.230","Text":"this is equal to 1."},{"Start":"04:45.230 ","End":"04:48.125","Text":"Essentially this is just the inverse fraction."},{"Start":"04:48.125 ","End":"04:52.400","Text":"In our case, in order to save the exercise and not to spoil it,"},{"Start":"04:52.400 ","End":"04:56.790","Text":"with this, If I now multiply by the opposite, in other words,"},{"Start":"04:56.790 ","End":"05:00.240","Text":"I\u0027ll put this on the numerator and here,"},{"Start":"05:00.240 ","End":"05:06.255","Text":"2 plus the square root of 3x plus 1,"},{"Start":"05:06.255 ","End":"05:09.980","Text":"then I\u0027ll be in a situation where I\u0027ve taken the original fraction,"},{"Start":"05:09.980 ","End":"05:14.330","Text":"multiplied it by sum a over b and then by sum b over a."},{"Start":"05:14.330 ","End":"05:17.360","Text":"This whole thing together is equal to 1,"},{"Start":"05:17.360 ","End":"05:19.975","Text":"so I haven\u0027t changed anything."},{"Start":"05:19.975 ","End":"05:27.090","Text":"This equals the limit as x goes to 1."},{"Start":"05:27.090 ","End":"05:31.729","Text":"Now, this times this we\u0027ve already done in this exercise,"},{"Start":"05:31.729 ","End":"05:34.510","Text":"and that\u0027s 3 minus 3x."},{"Start":"05:34.510 ","End":"05:39.500","Text":"This with this gives us what we did here,"},{"Start":"05:39.500 ","End":"05:43.260","Text":"which is 2 minus 2x."},{"Start":"05:44.510 ","End":"05:47.850","Text":"Now this times this is this,"},{"Start":"05:47.850 ","End":"05:49.260","Text":"I\u0027ll just put it in brackets,"},{"Start":"05:49.260 ","End":"05:51.855","Text":"the same thing with this pair in brackets."},{"Start":"05:51.855 ","End":"05:56.915","Text":"What we\u0027re left with is the numerator and denominator from the last of the 3,"},{"Start":"05:56.915 ","End":"06:04.335","Text":"which is brackets 1 plus the square root of 2x minus 1."},{"Start":"06:04.335 ","End":"06:12.720","Text":"Here, 2 plus the square root of 3x plus 1."},{"Start":"06:12.720 ","End":"06:16.530","Text":"Usually at this point, something cancels."},{"Start":"06:16.530 ","End":"06:19.025","Text":"We\u0027re looking for something to cancel."},{"Start":"06:19.025 ","End":"06:25.430","Text":"If you notice, we can take 3 out of this and 2 out of this,"},{"Start":"06:25.430 ","End":"06:31.100","Text":"and what we\u0027ll get is the limit x tends to 1,"},{"Start":"06:31.100 ","End":"06:36.860","Text":"3 times 1 minus x times the other piece,"},{"Start":"06:36.860 ","End":"06:38.165","Text":"I\u0027ll fill it in in a minute,"},{"Start":"06:38.165 ","End":"06:43.860","Text":"and here twice, 1 minus x times the last bit."},{"Start":"06:43.860 ","End":"06:48.690","Text":"1 plus the square root of 2x minus 1,"},{"Start":"06:48.690 ","End":"06:54.720","Text":"2 plus the square root of 3x plus 1."},{"Start":"06:54.720 ","End":"06:58.145","Text":"Now x tends to 1. x is not equal to 1,"},{"Start":"06:58.145 ","End":"07:00.440","Text":"so 1 minus x is not 0,"},{"Start":"07:00.440 ","End":"07:02.135","Text":"and we can cancel."},{"Start":"07:02.135 ","End":"07:07.220","Text":"After this cancellation, this function that\u0027s left is still elementary,"},{"Start":"07:07.220 ","End":"07:12.005","Text":"and this time we can substitute x equals 1 to get the answer."},{"Start":"07:12.005 ","End":"07:15.875","Text":"I\u0027m taking this expression and putting in x equals 1."},{"Start":"07:15.875 ","End":"07:18.070","Text":"What I get is"},{"Start":"07:18.070 ","End":"07:28.665","Text":"3 times 1 plus square root of 2 times 1 minus 1 over 2 from here,"},{"Start":"07:28.665 ","End":"07:31.500","Text":"2 plus, now x again is 1,"},{"Start":"07:31.500 ","End":"07:37.960","Text":"the square root of 3 times 1 plus 1."},{"Start":"07:38.710 ","End":"07:41.075","Text":"Let\u0027s see what we\u0027ve got here."},{"Start":"07:41.075 ","End":"07:45.360","Text":"This is 3, this is 2."},{"Start":"07:45.650 ","End":"07:49.410","Text":"Here we have 1 plus 1."},{"Start":"07:49.410 ","End":"07:55.065","Text":"Square root is 1, 2 plus 2."},{"Start":"07:55.065 ","End":"07:59.105","Text":"This is 6 over 8,"},{"Start":"07:59.105 ","End":"08:07.705","Text":"which we can simplify top and bottom divide by 2 to 3/4."},{"Start":"08:07.705 ","End":"08:10.540","Text":"That\u0027s the end of the exercise."}],"ID":1549},{"Watched":false,"Name":"Exercise 7","Duration":"4m 6s","ChapterTopicVideoID":1538,"CourseChapterTopicPlaylistID":164,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.915","Text":"In this exercise, we want to find the limit as x goes to 1 of the function,"},{"Start":"00:06.915 ","End":"00:10.680","Text":"1 minus the cube root of x over 1 minus x."},{"Start":"00:10.680 ","End":"00:17.730","Text":"Now, sometimes we can find the limit just by substituting x equals 1 into this function."},{"Start":"00:17.730 ","End":"00:20.775","Text":"But unfortunately it doesn\u0027t work here."},{"Start":"00:20.775 ","End":"00:22.620","Text":"Because if we put here x equals 1,"},{"Start":"00:22.620 ","End":"00:25.650","Text":"we get 0 and if we put x equals 1 here,"},{"Start":"00:25.650 ","End":"00:28.335","Text":"we also get 1 minus 1 is 0,"},{"Start":"00:28.335 ","End":"00:32.925","Text":"so this is one of those exercises of the form 0/0."},{"Start":"00:32.925 ","End":"00:35.400","Text":"So we\u0027re going to have to use some other techniques."},{"Start":"00:35.400 ","End":"00:38.835","Text":"Now, if this was a square root and not a cube root,"},{"Start":"00:38.835 ","End":"00:41.185","Text":"we\u0027d go with the method of the conjugates."},{"Start":"00:41.185 ","End":"00:45.320","Text":"There is something similar for cube roots as opposed to square roots."},{"Start":"00:45.320 ","End":"00:47.360","Text":"It\u0027s similar to conjugate,"},{"Start":"00:47.360 ","End":"00:51.670","Text":"and I\u0027d like to write some formulas from algebra for you."},{"Start":"00:51.670 ","End":"00:53.880","Text":"What I\u0027ve written is, first of all,"},{"Start":"00:53.880 ","End":"00:57.260","Text":"a reminder of what we used to do in the case of conjugates,"},{"Start":"00:57.260 ","End":"00:59.490","Text":"A minus B and A plus B are conjugates,"},{"Start":"00:59.490 ","End":"01:00.665","Text":"and if I multiply,"},{"Start":"01:00.665 ","End":"01:02.930","Text":"I get a difference of squares."},{"Start":"01:02.930 ","End":"01:06.890","Text":"There is a similar formula in algebra called a difference of cubes."},{"Start":"01:06.890 ","End":"01:09.500","Text":"Essentially, what I\u0027ve written here is that"},{"Start":"01:09.500 ","End":"01:12.230","Text":"if I have A minus B and multiply it by this thing,"},{"Start":"01:12.230 ","End":"01:14.030","Text":"A squared plus AB plus B squared,"},{"Start":"01:14.030 ","End":"01:15.515","Text":"I get a difference of cubes."},{"Start":"01:15.515 ","End":"01:18.545","Text":"This will help me to get rid of cube roots."},{"Start":"01:18.545 ","End":"01:21.455","Text":"Let\u0027s see how it works in our case."},{"Start":"01:21.455 ","End":"01:26.025","Text":"Look at the numerator first and think of this as my A minus B."},{"Start":"01:26.025 ","End":"01:30.105","Text":"So 1 minus the cube root of x."},{"Start":"01:30.105 ","End":"01:33.320","Text":"Instead of multiplying as we did in the square root case,"},{"Start":"01:33.320 ","End":"01:37.250","Text":"with the conjugate, I\u0027m going to multiply by the equivalent of what this is here,"},{"Start":"01:37.250 ","End":"01:40.465","Text":"so we get A squared,"},{"Start":"01:40.465 ","End":"01:43.335","Text":"which is 1 plus AB,"},{"Start":"01:43.335 ","End":"01:46.095","Text":"which is 1 times the cube root of x,"},{"Start":"01:46.095 ","End":"01:49.970","Text":"and B squared, which is the cube root of x squared."},{"Start":"01:49.970 ","End":"01:52.490","Text":"If I multiply this, what I get,"},{"Start":"01:52.490 ","End":"01:53.720","Text":"according to this formula,"},{"Start":"01:53.720 ","End":"01:55.820","Text":"is A cubed minus B cubed,"},{"Start":"01:55.820 ","End":"01:58.565","Text":"which is 1 cubed is 1,"},{"Start":"01:58.565 ","End":"02:02.660","Text":"and B cubed is the cube root cubed,"},{"Start":"02:02.660 ","End":"02:04.725","Text":"which is just the thing itself,"},{"Start":"02:04.725 ","End":"02:07.395","Text":"so I get 1 minus x."},{"Start":"02:07.395 ","End":"02:11.235","Text":"Let\u0027s see how this helps me in this exercise."},{"Start":"02:11.235 ","End":"02:17.450","Text":"I start off again by writing the limit and just copying of 1"},{"Start":"02:17.450 ","End":"02:25.714","Text":"minus the cube root of x over 1 minus x."},{"Start":"02:25.714 ","End":"02:31.190","Text":"What I\u0027d like to do is since multiplying by this gives such a simple expression,"},{"Start":"02:31.190 ","End":"02:36.560","Text":"I\u0027d like to multiply this by 1 plus"},{"Start":"02:36.560 ","End":"02:44.675","Text":"the cube root of x plus the cube root of x squared."},{"Start":"02:44.675 ","End":"02:48.320","Text":"I can\u0027t just go ahead and multiply because I\u0027m changing it."},{"Start":"02:48.320 ","End":"02:53.120","Text":"However, if I write the same thing on the denominator,"},{"Start":"02:53.120 ","End":"02:56.210","Text":"this fraction is of the form A/A,"},{"Start":"02:56.210 ","End":"02:58.415","Text":"same numerator as denominator,"},{"Start":"02:58.415 ","End":"03:00.770","Text":"and such a thing is equal to 1."},{"Start":"03:00.770 ","End":"03:03.950","Text":"I can certainly multiply by 1 without changing anything."},{"Start":"03:03.950 ","End":"03:06.740","Text":"Though continuing, do a bit of the algebra,"},{"Start":"03:06.740 ","End":"03:11.180","Text":"we get the limit as x goes to 1."},{"Start":"03:11.180 ","End":"03:13.010","Text":"This times this,"},{"Start":"03:13.010 ","End":"03:15.230","Text":"we\u0027ve already done over here, in other words,"},{"Start":"03:15.230 ","End":"03:19.070","Text":"we\u0027ve computed the numerator to be 1 minus x."},{"Start":"03:19.070 ","End":"03:21.545","Text":"Now in the denominator,"},{"Start":"03:21.545 ","End":"03:27.050","Text":"I just leave this 1 minus x as it is and copy this bit here."},{"Start":"03:27.050 ","End":"03:29.330","Text":"What I notice is 1 minus x here,"},{"Start":"03:29.330 ","End":"03:31.795","Text":"the 1 minus x here, and they cancel."},{"Start":"03:31.795 ","End":"03:34.905","Text":"Of course that leaves a 1 in the numerator."},{"Start":"03:34.905 ","End":"03:39.140","Text":"Now I\u0027ve got to find the limit as x goes to 1 of a much simpler function by"},{"Start":"03:39.140 ","End":"03:43.780","Text":"which I mean that we can substitute x equals 1 in this case where we couldn\u0027t before."},{"Start":"03:43.780 ","End":"03:48.155","Text":"All we have to do to evaluate this limit is to let x equal 1."},{"Start":"03:48.155 ","End":"03:58.165","Text":"We get just almost copying 1 over the 1 from here plus the cube root of 1 is 1,"},{"Start":"03:58.165 ","End":"04:01.590","Text":"and the cube root of 1 squared is also 1,"},{"Start":"04:01.590 ","End":"04:07.180","Text":"so the final answer is 1/3. That\u0027s it."}],"ID":1550}],"Thumbnail":null,"ID":164},{"Name":"Technique 4 Function Tends to Infinity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Function Tends to Infinity","Duration":"9m 43s","ChapterTopicVideoID":8251,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.505","Text":"In this clip, we introduce technique number 4 for finding limits of functions."},{"Start":"00:05.505 ","End":"00:08.775","Text":"This technique is functions that tend to infinity."},{"Start":"00:08.775 ","End":"00:11.700","Text":"At last, we talk about the number infinity,"},{"Start":"00:11.700 ","End":"00:13.740","Text":"which in fact is not even a number at all."},{"Start":"00:13.740 ","End":"00:19.350","Text":"It\u0027s a symbol for some quantity or number that grows without bound."},{"Start":"00:19.350 ","End":"00:22.110","Text":"In fact, there\u0027s also a minus infinity,"},{"Start":"00:22.110 ","End":"00:24.380","Text":"which is something which shrinks without bounds."},{"Start":"00:24.380 ","End":"00:27.140","Text":"When I say shrink, I mean get more and more negative,"},{"Start":"00:27.140 ","End":"00:28.350","Text":"not more and more tiny."},{"Start":"00:28.350 ","End":"00:30.090","Text":"When do we use such a technique?"},{"Start":"00:30.090 ","End":"00:35.805","Text":"When we have a limit of the form not 0 over tends to 0,"},{"Start":"00:35.805 ","End":"00:39.119","Text":"then we get a function that tends to infinity."},{"Start":"00:39.119 ","End":"00:40.690","Text":"I want to be more precise."},{"Start":"00:40.690 ","End":"00:44.690","Text":"This limit will be when x goes to some specific value,"},{"Start":"00:44.690 ","End":"00:52.205","Text":"say a and what I want is for this not 0 to be not 0 at x equals a,"},{"Start":"00:52.205 ","End":"00:56.555","Text":"and the tends to 0 as x tends to a,"},{"Start":"00:56.555 ","End":"00:58.220","Text":"just to be a bit more precise."},{"Start":"00:58.220 ","End":"01:00.590","Text":"This is a bit different from what we\u0027re familiar with."},{"Start":"01:00.590 ","End":"01:04.005","Text":"We\u0027re familiar with tends to 0 over tends to 0."},{"Start":"01:04.005 ","End":"01:07.295","Text":"Now we have not 0 over tends to 0."},{"Start":"01:07.295 ","End":"01:10.670","Text":"Let\u0027s do some examples to see what this really means."},{"Start":"01:10.670 ","End":"01:12.004","Text":"For the first example,"},{"Start":"01:12.004 ","End":"01:14.180","Text":"we\u0027ll take the simplest case;"},{"Start":"01:14.180 ","End":"01:20.015","Text":"the limit as x tends to 0 of 1 over x."},{"Start":"01:20.015 ","End":"01:23.195","Text":"Now here clearly the numerator is 1,"},{"Start":"01:23.195 ","End":"01:25.745","Text":"so it\u0027s definitely not 0 anywhere."},{"Start":"01:25.745 ","End":"01:30.560","Text":"The denominator x tends to 0 as x tends to 0."},{"Start":"01:30.560 ","End":"01:35.465","Text":"That\u0027s obvious. Here we have non 0 over tends to 0."},{"Start":"01:35.465 ","End":"01:44.670","Text":"Another example is the limit as x tends to 1 of x plus 2 over x minus 1."},{"Start":"01:44.670 ","End":"01:46.310","Text":"If we examined this again,"},{"Start":"01:46.310 ","End":"01:49.310","Text":"we\u0027ll see that technique number 4 is useful because"},{"Start":"01:49.310 ","End":"01:55.145","Text":"the numerator x plus 2 is definitely not 0 when x equals 1, it\u0027s 3."},{"Start":"01:55.145 ","End":"01:56.630","Text":"I\u0027m not saying it can\u0027t be 0."},{"Start":"01:56.630 ","End":"01:59.090","Text":"Sometimes it could be 0 when x is minus 2,"},{"Start":"01:59.090 ","End":"02:01.010","Text":"but we\u0027re not talking about minus 2,"},{"Start":"02:01.010 ","End":"02:02.090","Text":"we\u0027re talking about 1."},{"Start":"02:02.090 ","End":"02:03.470","Text":"Here it\u0027s definitely not 0."},{"Start":"02:03.470 ","End":"02:08.720","Text":"On the other hand, the denominator x minus 1 does go to 0 when x tends to 1."},{"Start":"02:08.720 ","End":"02:10.970","Text":"Quite clearly, x gets closer and closer to 1,"},{"Start":"02:10.970 ","End":"02:13.730","Text":"x minus 1 gets closer and closer to 0."},{"Start":"02:13.730 ","End":"02:15.650","Text":"This is a second example."},{"Start":"02:15.650 ","End":"02:17.915","Text":"Now let\u0027s look at a third example."},{"Start":"02:17.915 ","End":"02:23.050","Text":"Limit as x tends to 4 of x"},{"Start":"02:23.050 ","End":"02:28.670","Text":"squared plus x plus 1 over x minus 4."},{"Start":"02:28.670 ","End":"02:32.570","Text":"Once again, we look at the numerator to make sure it\u0027s not 0."},{"Start":"02:32.570 ","End":"02:33.920","Text":"If I put 4 in here,"},{"Start":"02:33.920 ","End":"02:36.770","Text":"4 squared plus 4 plus 1 is actually 21,"},{"Start":"02:36.770 ","End":"02:39.485","Text":"which is definitely not 0."},{"Start":"02:39.485 ","End":"02:42.680","Text":"Whereas the denominator definitely does go to"},{"Start":"02:42.680 ","End":"02:46.450","Text":"0 and something very important I have to add;"},{"Start":"02:46.450 ","End":"02:48.990","Text":"when we use this technique i.e.,"},{"Start":"02:48.990 ","End":"02:51.480","Text":"the not 0 over tends to 0,"},{"Start":"02:51.480 ","End":"02:56.990","Text":"it\u0027s very important to distinguish between the left limit and the right limit."},{"Start":"02:56.990 ","End":"02:58.580","Text":"This is because it makes"},{"Start":"02:58.580 ","End":"03:02.510","Text":"a very big difference if we\u0027re dividing by something that\u0027s very,"},{"Start":"03:02.510 ","End":"03:05.015","Text":"very close to 0 but positive,"},{"Start":"03:05.015 ","End":"03:08.555","Text":"or very, very close to 0 but negative."},{"Start":"03:08.555 ","End":"03:11.240","Text":"We\u0027ll see this in more detail later on."},{"Start":"03:11.240 ","End":"03:13.730","Text":"But first, let me put it down in writing."},{"Start":"03:13.730 ","End":"03:15.255","Text":"In such cases,"},{"Start":"03:15.255 ","End":"03:18.915","Text":"and I mean the non 0 over tends to 0,"},{"Start":"03:18.915 ","End":"03:22.520","Text":"we have to separate the limit into the limit from the left"},{"Start":"03:22.520 ","End":"03:26.150","Text":"or left limit and the limit from the right or the right limit."},{"Start":"03:26.150 ","End":"03:28.610","Text":"Only if these 2 are equal,"},{"Start":"03:28.610 ","End":"03:30.425","Text":"does the function have a limit."},{"Start":"03:30.425 ","End":"03:32.750","Text":"Again, we\u0027ll see this in the examples."},{"Start":"03:32.750 ","End":"03:35.315","Text":"Let\u0027s illustrate with the first example above,"},{"Start":"03:35.315 ","End":"03:42.455","Text":"which was the limit as x goes to 0 of 1 over x. I want to know what that is."},{"Start":"03:42.455 ","End":"03:47.270","Text":"I need the limit as x goes to 0 from the right,"},{"Start":"03:47.270 ","End":"03:52.010","Text":"you write it with a plus here and limit as x goes to 0"},{"Start":"03:52.010 ","End":"03:56.760","Text":"from the left of the same thing and I\u0027ll see what each of these is."},{"Start":"03:56.760 ","End":"03:58.610","Text":"I\u0027ve a bit of coloring here."},{"Start":"03:58.610 ","End":"03:59.810","Text":"Let\u0027s do the first one;"},{"Start":"03:59.810 ","End":"04:06.290","Text":"the x goes to 0 from the right and make a little table here of values x,"},{"Start":"04:06.290 ","End":"04:10.190","Text":"y, where y equals the 1 over x."},{"Start":"04:10.190 ","End":"04:14.060","Text":"If I take something like x equals something small,"},{"Start":"04:14.060 ","End":"04:15.665","Text":"1 over 100,"},{"Start":"04:15.665 ","End":"04:18.470","Text":"then y, which is 1 over x, will be 100."},{"Start":"04:18.470 ","End":"04:20.380","Text":"If I take x even smaller,"},{"Start":"04:20.380 ","End":"04:21.930","Text":"1 over, I don\u0027t know,"},{"Start":"04:21.930 ","End":"04:26.930","Text":"50,000, then y will be 50,000 getting larger."},{"Start":"04:26.930 ","End":"04:28.940","Text":"If I take something very, very small,"},{"Start":"04:28.940 ","End":"04:30.215","Text":"1 over 1,000,000,"},{"Start":"04:30.215 ","End":"04:32.060","Text":"then y is going to be 1,000,000."},{"Start":"04:32.060 ","End":"04:35.165","Text":"We can see that as x gets smaller and smaller,"},{"Start":"04:35.165 ","End":"04:37.729","Text":"but from the right in other words positive,"},{"Start":"04:37.729 ","End":"04:42.695","Text":"then y gets larger and larger without bound, it gets large."},{"Start":"04:42.695 ","End":"04:45.800","Text":"That\u0027s what we mean when we say goes to infinity."},{"Start":"04:45.800 ","End":"04:49.030","Text":"It gets large and is not bounded in size."},{"Start":"04:49.030 ","End":"04:52.775","Text":"Gets larger than anything you want and keep getting larger and larger."},{"Start":"04:52.775 ","End":"05:00.740","Text":"I\u0027m going to write in this case that the limit as x goes to 0 of 1 over x is infinity."},{"Start":"05:00.740 ","End":"05:05.405","Text":"Now how about when x goes to 0 from the left?"},{"Start":"05:05.405 ","End":"05:08.030","Text":"Which means that it\u0027s slightly negative."},{"Start":"05:08.030 ","End":"05:09.455","Text":"It gets closer and closer to 0,"},{"Start":"05:09.455 ","End":"05:11.300","Text":"but through negative numbers."},{"Start":"05:11.300 ","End":"05:13.355","Text":"I don\u0027t even have to make a new table."},{"Start":"05:13.355 ","End":"05:15.125","Text":"I could just put minus here,"},{"Start":"05:15.125 ","End":"05:17.150","Text":"minus here, and minus here."},{"Start":"05:17.150 ","End":"05:19.070","Text":"When x is minus 100th,"},{"Start":"05:19.070 ","End":"05:20.870","Text":"y is going to be minus 100."},{"Start":"05:20.870 ","End":"05:23.315","Text":"This looks like it\u0027s getting very, very small,"},{"Start":"05:23.315 ","End":"05:27.030","Text":"meaning negative, large but negative if you like."},{"Start":"05:27.030 ","End":"05:30.200","Text":"The answer here is going to be minus infinity."},{"Start":"05:30.200 ","End":"05:34.010","Text":"Now what\u0027s happened is that there is a limit on the left,"},{"Start":"05:34.010 ","End":"05:36.770","Text":"at least in the incentive infinity or minus infinity,"},{"Start":"05:36.770 ","End":"05:38.465","Text":"but they\u0027re not equal."},{"Start":"05:38.465 ","End":"05:45.470","Text":"What I say in such a case is that the limit of 1 over x as a whole does not exist."},{"Start":"05:45.470 ","End":"05:49.010","Text":"Of course, if both of these had come out infinity,"},{"Start":"05:49.010 ","End":"05:51.620","Text":"then I would have written this limit as infinity,"},{"Start":"05:51.620 ","End":"05:53.960","Text":"or if both of these would come out minus infinity l"},{"Start":"05:53.960 ","End":"05:56.725","Text":"also would have written this limit as minus infinity."},{"Start":"05:56.725 ","End":"05:58.320","Text":"But here they didn\u0027t come out the same,"},{"Start":"05:58.320 ","End":"05:59.760","Text":"so there just is no limit."},{"Start":"05:59.760 ","End":"06:02.120","Text":"I would like to show you what happens graphically,"},{"Start":"06:02.120 ","End":"06:04.175","Text":"just to help you get an idea."},{"Start":"06:04.175 ","End":"06:05.900","Text":"Y equals 1 over x,"},{"Start":"06:05.900 ","End":"06:08.345","Text":"which consists of 2 parts."},{"Start":"06:08.345 ","End":"06:11.060","Text":"This part here and this part here."},{"Start":"06:11.060 ","End":"06:17.090","Text":"The idea is I take values that keep getting closer and closer to 0,"},{"Start":"06:17.090 ","End":"06:19.010","Text":"but from the positive side,"},{"Start":"06:19.010 ","End":"06:21.305","Text":"then above them, on the graph,"},{"Start":"06:21.305 ","End":"06:24.560","Text":"I\u0027m going to get values that are just going to"},{"Start":"06:24.560 ","End":"06:28.750","Text":"get larger and larger and larger towards infinity."},{"Start":"06:28.750 ","End":"06:32.255","Text":"If we take values that go to 0 from the left,"},{"Start":"06:32.255 ","End":"06:36.035","Text":"we\u0027ll get values that go to minus infinity."},{"Start":"06:36.035 ","End":"06:39.215","Text":"These 2 don\u0027t meet and it\u0027s not the same limit."},{"Start":"06:39.215 ","End":"06:40.580","Text":"Just by the way,"},{"Start":"06:40.580 ","End":"06:42.800","Text":"if you have a line such as the y-axis,"},{"Start":"06:42.800 ","End":"06:45.740","Text":"in this case, which the function tends towards,"},{"Start":"06:45.740 ","End":"06:49.180","Text":"gets closer and closer but never actually reaches,"},{"Start":"06:49.180 ","End":"06:51.900","Text":"then this line, vertical line like this,"},{"Start":"06:51.900 ","End":"06:54.045","Text":"is called a vertical asymptote."},{"Start":"06:54.045 ","End":"06:55.925","Text":"Onto another example."},{"Start":"06:55.925 ","End":"06:58.520","Text":"This time we\u0027ll take as an example,"},{"Start":"06:58.520 ","End":"07:01.250","Text":"the limit as x goes to 0,"},{"Start":"07:01.250 ","End":"07:05.480","Text":"not 1 over x, but this time 1 over x squared."},{"Start":"07:05.480 ","End":"07:11.565","Text":"As before, this is something which is not 0 over something that tends to 0."},{"Start":"07:11.565 ","End":"07:13.805","Text":"We split it up into 2 cases."},{"Start":"07:13.805 ","End":"07:18.890","Text":"We take the limit as x goes to 0 from the right,"},{"Start":"07:18.890 ","End":"07:20.450","Text":"I\u0027ll write that in a minute,"},{"Start":"07:20.450 ","End":"07:25.430","Text":"of 1 over x squared and the limit as x goes to 0 from the left,"},{"Start":"07:25.430 ","End":"07:26.795","Text":"I\u0027ll put that in a minute,"},{"Start":"07:26.795 ","End":"07:28.970","Text":"of 1 over x squared."},{"Start":"07:28.970 ","End":"07:31.265","Text":"I want to put a plus in blue,"},{"Start":"07:31.265 ","End":"07:33.935","Text":"but a minus in red."},{"Start":"07:33.935 ","End":"07:36.370","Text":"Just like before, we made a table."},{"Start":"07:36.370 ","End":"07:41.420","Text":"If we start out with some simple examples like x is something small,"},{"Start":"07:41.420 ","End":"07:43.940","Text":"maybe 1 over 100,"},{"Start":"07:43.940 ","End":"07:46.930","Text":"then we\u0027ll try 1 over 1,000,"},{"Start":"07:46.930 ","End":"07:50.550","Text":"then we\u0027ll try 1 over 10,000."},{"Start":"07:50.550 ","End":"07:54.425","Text":"We get the corresponding y is 1 over x squared,"},{"Start":"07:54.425 ","End":"07:56.240","Text":"so it\u0027s 1 over 100 squared."},{"Start":"07:56.240 ","End":"07:58.775","Text":"This is actually 10,000."},{"Start":"07:58.775 ","End":"08:06.010","Text":"1 over 1,000 is 1,000,000 and 1 over 10,000 is 100,000,000."},{"Start":"08:06.010 ","End":"08:08.885","Text":"This also gets large without bound,"},{"Start":"08:08.885 ","End":"08:10.930","Text":"but much quicker than the one before."},{"Start":"08:10.930 ","End":"08:15.255","Text":"We can easily see that this one is infinity."},{"Start":"08:15.255 ","End":"08:19.085","Text":"If we take the other limit on the left,"},{"Start":"08:19.085 ","End":"08:23.120","Text":"then all I have to do is add a minus in front of these."},{"Start":"08:23.120 ","End":"08:27.800","Text":"But notice that these don\u0027t change because also minus 100,"},{"Start":"08:27.800 ","End":"08:29.090","Text":"if I take 1 over x squared,"},{"Start":"08:29.090 ","End":"08:32.585","Text":"it\u0027s also 10,000 plus because of the squared."},{"Start":"08:32.585 ","End":"08:36.980","Text":"Here we also get infinity and that means that this"},{"Start":"08:36.980 ","End":"08:41.600","Text":"time we do have a limit here and this is equal to infinity."},{"Start":"08:41.600 ","End":"08:45.380","Text":"As before, I\u0027d also like to show you what happens graphically."},{"Start":"08:45.380 ","End":"08:48.230","Text":"I\u0027ll just do a real rough and ready sketch here."},{"Start":"08:48.230 ","End":"08:52.985","Text":"There\u0027s not going to be anything below the x-axis because everything\u0027s positive."},{"Start":"08:52.985 ","End":"08:55.280","Text":"This one goes something like this,"},{"Start":"08:55.280 ","End":"08:58.880","Text":"I happen to know, and this one goes also just the mirror image of it."},{"Start":"08:58.880 ","End":"09:00.955","Text":"Notice that it\u0027s an even function."},{"Start":"09:00.955 ","End":"09:05.165","Text":"When we go to 0 from the positive side,"},{"Start":"09:05.165 ","End":"09:07.370","Text":"then we\u0027re going to infinity."},{"Start":"09:07.370 ","End":"09:11.080","Text":"In other words, the numbers keep getting larger and larger without bound."},{"Start":"09:11.080 ","End":"09:14.825","Text":"On the other side, when x go to 0 from the left,"},{"Start":"09:14.825 ","End":"09:19.000","Text":"the values get larger and larger with no limit."},{"Start":"09:19.000 ","End":"09:25.565","Text":"In other words, this side is the minus infinity and this side it goes to plus infinity,"},{"Start":"09:25.565 ","End":"09:27.515","Text":"sorry, this is also infinity."},{"Start":"09:27.515 ","End":"09:30.260","Text":"You\u0027ll see plenty more examples like this in"},{"Start":"09:30.260 ","End":"09:33.695","Text":"the exercises following the theoretical section."},{"Start":"09:33.695 ","End":"09:36.860","Text":"As in the previous sketch with 1 over x,"},{"Start":"09:36.860 ","End":"09:40.955","Text":"the y axis itself is something called an asymptote to this function;"},{"Start":"09:40.955 ","End":"09:43.500","Text":"y equals 1 over x squared."}],"ID":8411},{"Watched":false,"Name":"Exercise 1","Duration":"2m 5s","ChapterTopicVideoID":4739,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.780","Text":"In this exercise, we have to find the limit as x goes to 0 of x squared plus 4 over x."},{"Start":"00:06.780 ","End":"00:10.650","Text":"Now, if we try substituting x is equal to 0,"},{"Start":"00:10.650 ","End":"00:15.510","Text":"what we\u0027re going to get is that the denominator here is 0,"},{"Start":"00:15.510 ","End":"00:19.875","Text":"but the numerator is 0 squared plus 4 is not 0."},{"Start":"00:19.875 ","End":"00:21.149","Text":"Now, in general,"},{"Start":"00:21.149 ","End":"00:24.615","Text":"when we have the case where we have something of the form"},{"Start":"00:24.615 ","End":"00:28.260","Text":"not 0 over something that is 0,"},{"Start":"00:28.260 ","End":"00:33.275","Text":"then we have to separate the limit into the left limit and the right limit."},{"Start":"00:33.275 ","End":"00:37.130","Text":"Otherwise, we have to figure out what is the limit as x"},{"Start":"00:37.130 ","End":"00:41.180","Text":"goes to 0 from the left of the same thing,"},{"Start":"00:41.180 ","End":"00:46.745","Text":"and also the limit as x goes to 0 from the right,"},{"Start":"00:46.745 ","End":"00:48.890","Text":"which we write it like this."},{"Start":"00:48.890 ","End":"00:53.665","Text":"X squared plus 4_over_x."},{"Start":"00:53.665 ","End":"00:55.265","Text":"Let\u0027s see what this is."},{"Start":"00:55.265 ","End":"00:58.370","Text":"If we substitute 0 minus,"},{"Start":"00:58.370 ","End":"01:02.750","Text":"it\u0027s shorthand symbol for infinitesimally small amount,"},{"Start":"01:02.750 ","End":"01:05.300","Text":"close to 0 but just negative."},{"Start":"01:05.300 ","End":"01:09.690","Text":"So 0_squared plus 4 is 4 in this expression,"},{"Start":"01:09.690 ","End":"01:13.110","Text":"but when it\u0027s on its own we have to leave it as 0 minus."},{"Start":"01:13.110 ","End":"01:16.760","Text":"Which means a tiny bit and infinitesimal amount smaller than 0,"},{"Start":"01:16.760 ","End":"01:20.240","Text":"so it\u0027s 4 over something very small and negative,"},{"Start":"01:20.240 ","End":"01:24.035","Text":"which means it\u0027s going to be very large and negative"},{"Start":"01:24.035 ","End":"01:27.905","Text":"like 4 over minus 1/1,000,000 would be 4,000,000."},{"Start":"01:27.905 ","End":"01:31.080","Text":"This is equal to minus infinity, which means,"},{"Start":"01:31.080 ","End":"01:33.320","Text":"minus a very large number,"},{"Start":"01:33.320 ","End":"01:35.950","Text":"as large as you want by making this as small as you want."},{"Start":"01:35.950 ","End":"01:37.025","Text":"On the other side,"},{"Start":"01:37.025 ","End":"01:41.000","Text":"when x goes to 0 from the right means it\u0027s slightly positive,"},{"Start":"01:41.000 ","End":"01:43.610","Text":"so 0 squared plus 4 is 4,"},{"Start":"01:43.610 ","End":"01:47.840","Text":"but here we have positive 0 or positive infinitesimal amount,"},{"Start":"01:47.840 ","End":"01:49.745","Text":"and this is very large,"},{"Start":"01:49.745 ","End":"01:51.020","Text":"but positive this time."},{"Start":"01:51.020 ","End":"01:54.375","Text":"This equals just for emphasis plus infinity."},{"Start":"01:54.375 ","End":"01:58.594","Text":"Since we have here minus infinity and here we have plus infinity,"},{"Start":"01:58.594 ","End":"02:00.844","Text":"and these 2 are not equal,"},{"Start":"02:00.844 ","End":"02:04.220","Text":"then the answer is that there is no limit."},{"Start":"02:04.220 ","End":"02:06.450","Text":"It\u0027s undefined."}],"ID":4748},{"Watched":false,"Name":"Exercise 2","Duration":"1m 37s","ChapterTopicVideoID":4740,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this exercise, we have to find the limit as x"},{"Start":"00:02.850 ","End":"00:06.390","Text":"goes to 2 of x minus 1 squared over x minus 2."},{"Start":"00:06.390 ","End":"00:09.120","Text":"Let\u0027s see what happens if we substitute x equals 2."},{"Start":"00:09.120 ","End":"00:11.924","Text":"In the denominator 2 minus 2 is 0."},{"Start":"00:11.924 ","End":"00:14.775","Text":"Here, 2 minus 1 is 1 squared is 1,"},{"Start":"00:14.775 ","End":"00:16.320","Text":"so we get 1 over 0."},{"Start":"00:16.320 ","End":"00:22.530","Text":"The essential thing is that we get something which is non-zero over 0."},{"Start":"00:22.530 ","End":"00:25.005","Text":"Now in the case of non-zero over 0,"},{"Start":"00:25.005 ","End":"00:27.090","Text":"what we have to do is separately compute"},{"Start":"00:27.090 ","End":"00:29.520","Text":"the limit from the right and the limit from the left."},{"Start":"00:29.520 ","End":"00:32.640","Text":"In other words, we have to compute the limit as x"},{"Start":"00:32.640 ","End":"00:36.395","Text":"goes to 2 from the right of same thing,"},{"Start":"00:36.395 ","End":"00:42.620","Text":"and the limit as x goes to 2 from the left of the same thing."},{"Start":"00:42.620 ","End":"00:45.530","Text":"Substituting 2 plus we get,"},{"Start":"00:45.530 ","End":"00:48.275","Text":"2 minus 1 squared is 4,"},{"Start":"00:48.275 ","End":"00:55.480","Text":"2 plus minus 2 is what we call 0 plus a symbol for infinitesimally small but positive,"},{"Start":"00:55.480 ","End":"00:58.790","Text":"and this is equal to plus infinity."},{"Start":"00:58.790 ","End":"01:01.850","Text":"Whereas here we get 4 over"},{"Start":"01:01.850 ","End":"01:06.860","Text":"infinitesimally small but negative and that\u0027s equal to minus infinity."},{"Start":"01:06.860 ","End":"01:14.915","Text":"There is general formula that says that a/0 plus is equal to plus infinity."},{"Start":"01:14.915 ","End":"01:19.040","Text":"All this is in the case where if a is a positive quantity,"},{"Start":"01:19.040 ","End":"01:23.280","Text":"and a/0 minus is minus infinity."},{"Start":"01:23.280 ","End":"01:24.305","Text":"I\u0027ve used that here."},{"Start":"01:24.305 ","End":"01:26.600","Text":"The point to note is the right limit,"},{"Start":"01:26.600 ","End":"01:28.115","Text":"the left limit are not equal."},{"Start":"01:28.115 ","End":"01:30.785","Text":"These two things are not equal,"},{"Start":"01:30.785 ","End":"01:33.275","Text":"and when the right limit and the left limit are not equal,"},{"Start":"01:33.275 ","End":"01:35.480","Text":"then this limit doesn\u0027t exist."},{"Start":"01:35.480 ","End":"01:38.130","Text":"There is no limit."}],"ID":4749},{"Watched":false,"Name":"Exercise 3","Duration":"2m 13s","ChapterTopicVideoID":4741,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we have to find the limit as x tends to"},{"Start":"00:03.300 ","End":"00:07.440","Text":"2 of x squared minus 1 over x minus 2, x minus 5."},{"Start":"00:07.440 ","End":"00:11.790","Text":"The first thing to try is to substitute x equals 2 and see if there\u0027s a problem."},{"Start":"00:11.790 ","End":"00:13.845","Text":"What we get on the denominator,"},{"Start":"00:13.845 ","End":"00:15.600","Text":"if x goes to 2, this is 0,"},{"Start":"00:15.600 ","End":"00:18.090","Text":"so the denominator 0, but the numerator,"},{"Start":"00:18.090 ","End":"00:21.435","Text":"2 squared minus 1 is 3 is not 0."},{"Start":"00:21.435 ","End":"00:28.274","Text":"What this is, this is something of the form non-zero over something which is 0."},{"Start":"00:28.274 ","End":"00:32.115","Text":"Or I could write it as not 0/0."},{"Start":"00:32.115 ","End":"00:34.890","Text":"In this case, what we have to do is compute"},{"Start":"00:34.890 ","End":"00:38.480","Text":"separately the left limit and the right limit and if they\u0027re equal,"},{"Start":"00:38.480 ","End":"00:40.640","Text":"then we have a limit and otherwise not."},{"Start":"00:40.640 ","End":"00:47.540","Text":"Let\u0027s look at the limit as x goes to 2 from the right of the same thing."},{"Start":"00:47.540 ","End":"00:54.340","Text":"We\u0027re going to separately look at the limit as x goes to 2 from the left."},{"Start":"00:54.340 ","End":"00:55.545","Text":"Now, in this case,"},{"Start":"00:55.545 ","End":"00:57.930","Text":"if x is 2 from the right,"},{"Start":"00:57.930 ","End":"01:01.530","Text":"then we get 2 plus minus 2 is 0 plus"},{"Start":"01:01.530 ","End":"01:05.190","Text":"which mean something infinitesimally small but positive,"},{"Start":"01:05.190 ","End":"01:07.390","Text":"so this is 0 plus,"},{"Start":"01:07.390 ","End":"01:10.610","Text":"2 minus 5 is minus 3."},{"Start":"01:10.610 ","End":"01:13.880","Text":"We don\u0027t have to write pluses and minuses except when it\u0027s 0."},{"Start":"01:13.880 ","End":"01:18.320","Text":"On the numerator, 2 squared minus 1 is 3."},{"Start":"01:18.320 ","End":"01:25.940","Text":"What we get is the denominator 0 plus times a negative quantity is 0 minus,"},{"Start":"01:25.940 ","End":"01:28.475","Text":"the one of those rules of these infinitesimals,"},{"Start":"01:28.475 ","End":"01:29.900","Text":"something very, very tiny,"},{"Start":"01:29.900 ","End":"01:32.660","Text":"but positive times a negative will be very tiny,"},{"Start":"01:32.660 ","End":"01:35.750","Text":"even negative, 0 minus, this is 3."},{"Start":"01:35.750 ","End":"01:40.145","Text":"The positive over 0 minus is minus infinity."},{"Start":"01:40.145 ","End":"01:43.415","Text":"Here we get something very similar, 3,"},{"Start":"01:43.415 ","End":"01:50.300","Text":"only when x goes to 2 minus 2 minus less 2 is something close to 0 but slightly negative."},{"Start":"01:50.300 ","End":"01:52.670","Text":"The same thing here as minus 3,"},{"Start":"01:52.670 ","End":"01:58.595","Text":"as negative 0 times a negative number equals slightly positive 0."},{"Start":"01:58.595 ","End":"01:59.735","Text":"Again, the 3."},{"Start":"01:59.735 ","End":"02:03.170","Text":"Positive over 0 plus, is plus infinity."},{"Start":"02:03.170 ","End":"02:06.335","Text":"The thing is that these 2 are not equal,"},{"Start":"02:06.335 ","End":"02:07.400","Text":"and if they\u0027re not equal,"},{"Start":"02:07.400 ","End":"02:10.385","Text":"then this thing doesn\u0027t have a limit, it doesn\u0027t exist."},{"Start":"02:10.385 ","End":"02:11.780","Text":"No limit."},{"Start":"02:11.780 ","End":"02:13.950","Text":"That is the answer."}],"ID":4750},{"Watched":false,"Name":"Exercise 4","Duration":"1m 21s","ChapterTopicVideoID":4742,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, we want to find the limit as x goes to 0"},{"Start":"00:03.750 ","End":"00:07.715","Text":"from the right of natural log of x/x."},{"Start":"00:07.715 ","End":"00:10.895","Text":"The first thing we try to do is to substitute."},{"Start":"00:10.895 ","End":"00:14.235","Text":"If we substitute x goes to 0 from the right,"},{"Start":"00:14.235 ","End":"00:18.260","Text":"then x is just 0 from the right, 0 plus."},{"Start":"00:18.260 ","End":"00:22.505","Text":"But what happens when x goes to 0 from the right of the natural logarithm?"},{"Start":"00:22.505 ","End":"00:25.470","Text":"This is actually equal to minus infinity,"},{"Start":"00:25.470 ","End":"00:28.980","Text":"and the way I usually remember this is just from the graph,"},{"Start":"00:28.980 ","End":"00:30.450","Text":"the natural logarithm,"},{"Start":"00:30.450 ","End":"00:33.030","Text":"it\u0027s only defined for x is bigger than 0,"},{"Start":"00:33.030 ","End":"00:35.490","Text":"so x could only go from the right and x is 1,"},{"Start":"00:35.490 ","End":"00:37.290","Text":"it\u0027s equal to 0,"},{"Start":"00:37.290 ","End":"00:40.080","Text":"and then it goes down asymptotically."},{"Start":"00:40.080 ","End":"00:41.565","Text":"In this area here,"},{"Start":"00:41.565 ","End":"00:43.805","Text":"when x goes to 0,"},{"Start":"00:43.805 ","End":"00:47.465","Text":"y goes down all the way to minus infinity."},{"Start":"00:47.465 ","End":"00:53.705","Text":"What we would say is the natural log of 0 plus is minus infinity."},{"Start":"00:53.705 ","End":"00:58.595","Text":"Back to here, this minus infinity over the 0 plus,"},{"Start":"00:58.595 ","End":"01:05.315","Text":"I could write it as minus infinity times 1/0 plus."},{"Start":"01:05.315 ","End":"01:08.880","Text":"Now a positive number over 0 plus is plus infinity,"},{"Start":"01:08.880 ","End":"01:12.815","Text":"so we have minus infinity times plus infinity."},{"Start":"01:12.815 ","End":"01:14.450","Text":"The minus and the plus give us"},{"Start":"01:14.450 ","End":"01:22.410","Text":"a minus and infinity times infinity is just infinity so the answer is minus infinity."}],"ID":4751},{"Watched":false,"Name":"Exercise 5","Duration":"1m 47s","ChapterTopicVideoID":4816,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.430","Text":"In this exercise, we have to find the limit of x goes to 2_minus."},{"Start":"00:04.430 ","End":"00:05.460","Text":"Now, what does this mean?"},{"Start":"00:05.460 ","End":"00:07.935","Text":"This means that x goes to 2 from the left,"},{"Start":"00:07.935 ","End":"00:12.820","Text":"the left limit of minus 1/2 natural log of 2 minus x ."},{"Start":"00:12.820 ","End":"00:14.699","Text":"In this kind of exercise,"},{"Start":"00:14.699 ","End":"00:18.600","Text":"what we do is we just substitute x is equal to 2_minus,"},{"Start":"00:18.600 ","End":"00:19.965","Text":"which is something symbolic,"},{"Start":"00:19.965 ","End":"00:27.060","Text":"but actually works because 2 less 2_minus is 0_plus,"},{"Start":"00:27.060 ","End":"00:29.955","Text":"and if this confuses you can just think at the side and say,"},{"Start":"00:29.955 ","End":"00:36.150","Text":"something like 2.000 less 1.999 and so"},{"Start":"00:36.150 ","End":"00:43.175","Text":"on and this is going to equal something like 0.001 but it could be many 0s."},{"Start":"00:43.175 ","End":"00:47.900","Text":"Something very small but positive and that\u0027s where we get this equation from."},{"Start":"00:47.900 ","End":"00:51.050","Text":"Now, if we substitute the 2 minus for x,"},{"Start":"00:51.050 ","End":"00:58.520","Text":"we get that this thing is just minus 1/2 natural log of 0_plus."},{"Start":"00:58.520 ","End":"01:02.675","Text":"Now, this natural log of 0 plus is also something you should remember,"},{"Start":"01:02.675 ","End":"01:04.040","Text":"but I\u0027m going to remind you,"},{"Start":"01:04.040 ","End":"01:05.690","Text":"this is minus infinity,"},{"Start":"01:05.690 ","End":"01:10.130","Text":"this is minus 1/2 times minus infinity and I\u0027ll just"},{"Start":"01:10.130 ","End":"01:15.115","Text":"show you quickly why this is so because if we just take a quick graph here,"},{"Start":"01:15.115 ","End":"01:16.285","Text":"1 at 0,"},{"Start":"01:16.285 ","End":"01:19.490","Text":"and this is the x-axis of course, and the y-axis."},{"Start":"01:19.490 ","End":"01:22.280","Text":"When x goes to 0 from the right,"},{"Start":"01:22.280 ","End":"01:23.720","Text":"that\u0027s what the 0_plus means,"},{"Start":"01:23.720 ","End":"01:26.015","Text":"it\u0027s like it\u0027s going here and here and here,"},{"Start":"01:26.015 ","End":"01:30.005","Text":"the function goes lower and lower and lower down to minus infinity."},{"Start":"01:30.005 ","End":"01:31.660","Text":"That\u0027s where we get this from."},{"Start":"01:31.660 ","End":"01:35.240","Text":"Now, this is equal to minus times minus is plus,"},{"Start":"01:35.240 ","End":"01:37.910","Text":"so it\u0027s 1/2 times infinity,"},{"Start":"01:37.910 ","End":"01:44.510","Text":"and anything positive times infinity is just equal to infinity because it was negative,"},{"Start":"01:44.510 ","End":"01:48.250","Text":"it would be minus infinity and that\u0027s our answer."}],"ID":4816},{"Watched":false,"Name":"Exercise 6","Duration":"2m 25s","ChapterTopicVideoID":4817,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.780","Text":"In this exercise, we have to find the limit as x goes to 0 plus."},{"Start":"00:04.780 ","End":"00:07.805","Text":"Remember that means x goes to 0 from the right,"},{"Start":"00:07.805 ","End":"00:09.650","Text":"0 plus is just a symbol."},{"Start":"00:09.650 ","End":"00:17.040","Text":"What we do in such cases is we just substitute 0 plus as if it were an actual quantity."},{"Start":"00:17.040 ","End":"00:18.720","Text":"If we do that,"},{"Start":"00:18.720 ","End":"00:20.565","Text":"what we\u0027re going to get here,"},{"Start":"00:20.565 ","End":"00:23.475","Text":"and I have to remind you of a formula that"},{"Start":"00:23.475 ","End":"00:30.405","Text":"the natural log of 0 plus is equal to minus infinity."},{"Start":"00:30.405 ","End":"00:35.680","Text":"Here we would get natural log of 0 plus is minus infinity,"},{"Start":"00:35.680 ","End":"00:44.420","Text":"we would get minus infinity squared plus twice minus infinity minus 3."},{"Start":"00:44.420 ","End":"00:47.990","Text":"Now, this thing is infinity plus infinity,"},{"Start":"00:47.990 ","End":"00:53.930","Text":"twice minus infinity is minus infinity minus 3."},{"Start":"00:53.930 ","End":"00:56.480","Text":"The problem is with this thing,"},{"Start":"00:56.480 ","End":"01:00.530","Text":"infinity minus infinity is 1 of those forms which are called indeterminate,"},{"Start":"01:00.530 ","End":"01:02.150","Text":"you can\u0027t say what it is,"},{"Start":"01:02.150 ","End":"01:05.075","Text":"in different exercises that could be different things."},{"Start":"01:05.075 ","End":"01:07.640","Text":"This approach is not going to work."},{"Start":"01:07.640 ","End":"01:08.750","Text":"What are we going to do?"},{"Start":"01:08.750 ","End":"01:11.240","Text":"Well, let\u0027s try some algebra here."},{"Start":"01:11.240 ","End":"01:14.270","Text":"What we\u0027ll do is rewrite this."},{"Start":"01:14.270 ","End":"01:18.450","Text":"We\u0027ll have the limit x goes to 0 plus."},{"Start":"01:18.450 ","End":"01:21.320","Text":"I\u0027ll take this thing and take natural log of"},{"Start":"01:21.320 ","End":"01:24.560","Text":"x outside the brackets from the first 2 terms."},{"Start":"01:24.560 ","End":"01:26.885","Text":"We\u0027ll get x goes to 0,"},{"Start":"01:26.885 ","End":"01:29.224","Text":"natural log of x,"},{"Start":"01:29.224 ","End":"01:34.670","Text":"brackets, natural log of x plus 2,"},{"Start":"01:34.670 ","End":"01:36.620","Text":"and all this minus 3."},{"Start":"01:36.620 ","End":"01:37.715","Text":"If we do this,"},{"Start":"01:37.715 ","End":"01:41.810","Text":"now we\u0027ll get better results if we put the 0 plus here,"},{"Start":"01:41.810 ","End":"01:49.935","Text":"because here we get minus infinity times minus infinity plus 2,"},{"Start":"01:49.935 ","End":"01:52.610","Text":"and all this minus 3."},{"Start":"01:52.610 ","End":"01:55.920","Text":"Now this equals minus infinity,"},{"Start":"01:55.920 ","End":"01:58.475","Text":"that whenever you have plus or minus infinity,"},{"Start":"01:58.475 ","End":"02:02.520","Text":"adding or subtracting a finite quantity doesn\u0027t change it."},{"Start":"02:02.520 ","End":"02:05.660","Text":"This is still minus infinity,"},{"Start":"02:05.660 ","End":"02:10.800","Text":"minus 3, minus infinity times minus infinity is plus infinity."},{"Start":"02:10.800 ","End":"02:14.565","Text":"So we get infinity minus 3."},{"Start":"02:14.565 ","End":"02:19.684","Text":"As I said, if you subtract finite quantity from infinity or minus infinity,"},{"Start":"02:19.684 ","End":"02:21.050","Text":"it doesn\u0027t change it,"},{"Start":"02:21.050 ","End":"02:24.350","Text":"so this is just equal to infinity."},{"Start":"02:24.350 ","End":"02:26.550","Text":"That\u0027s the answer."}],"ID":4817},{"Watched":false,"Name":"Exercise 7","Duration":"2m 39s","ChapterTopicVideoID":4818,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.345","Text":"In this exercise, we have to find the limit as x goes to 0 of e to the power of 1 over x."},{"Start":"00:06.345 ","End":"00:09.390","Text":"This is 1 of those cases that will be just can\u0027t substitute x equals"},{"Start":"00:09.390 ","End":"00:12.840","Text":"0 because 1 over 0 is not defined."},{"Start":"00:12.840 ","End":"00:15.630","Text":"What we do in these cases is we"},{"Start":"00:15.630 ","End":"00:19.815","Text":"separate the limit into the left limit and the right limit."},{"Start":"00:19.815 ","End":"00:24.780","Text":"In other words, what I\u0027m going to have to compute is the limit as"},{"Start":"00:24.780 ","End":"00:30.810","Text":"x goes to 0 plus of e to the power of 1 over x."},{"Start":"00:30.810 ","End":"00:39.540","Text":"Also a limit as x goes to 0 from the left of e to the power of 1 over x."},{"Start":"00:39.540 ","End":"00:42.190","Text":"We\u0027ll see what each of these is equal to."},{"Start":"00:42.190 ","End":"00:47.540","Text":"I\u0027d like to give you a small review of some properties of the exponential function."},{"Start":"00:47.540 ","End":"00:53.690","Text":"I could just write this as a formula that e to the power of infinity is equal"},{"Start":"00:53.690 ","End":"00:59.870","Text":"to infinity and e to the power of minus infinity is 0,"},{"Start":"00:59.870 ","End":"01:01.800","Text":"and you can either just remember these."},{"Start":"01:01.800 ","End":"01:04.925","Text":"Or the simplest thing is what I do is I remember"},{"Start":"01:04.925 ","End":"01:08.695","Text":"the rough shape of the graph of e to the power of x."},{"Start":"01:08.695 ","End":"01:11.840","Text":"Where this is the x-axis,"},{"Start":"01:11.840 ","End":"01:13.910","Text":"this is the y-axis,"},{"Start":"01:13.910 ","End":"01:15.590","Text":"and e to the power of x,"},{"Start":"01:15.590 ","End":"01:17.980","Text":"it goes through the 0.01."},{"Start":"01:17.980 ","End":"01:24.005","Text":"But on this side it goes down to 0 asymptotically and in here it goes to infinity."},{"Start":"01:24.005 ","End":"01:26.285","Text":"Other words, in this direction,"},{"Start":"01:26.285 ","End":"01:28.345","Text":"the graph goes down to 0,"},{"Start":"01:28.345 ","End":"01:30.320","Text":"and in this direction, the graph,"},{"Start":"01:30.320 ","End":"01:32.510","Text":"meaning y goes up to infinity."},{"Start":"01:32.510 ","End":"01:35.120","Text":"When x goes to infinity,"},{"Start":"01:35.120 ","End":"01:36.814","Text":"e to the infinity is infinity,"},{"Start":"01:36.814 ","End":"01:39.020","Text":"and when x goes to minus infinity,"},{"Start":"01:39.020 ","End":"01:41.465","Text":"e to minus infinity is 0."},{"Start":"01:41.465 ","End":"01:43.535","Text":"Let\u0027s see how we use that here."},{"Start":"01:43.535 ","End":"01:48.515","Text":"If x goes to 0 plus then e to the 1 over x,"},{"Start":"01:48.515 ","End":"01:53.805","Text":"we get e to the power of 1 over 0 plus."},{"Start":"01:53.805 ","End":"01:57.280","Text":"Now 1 over 0 plus is plus infinity."},{"Start":"01:57.280 ","End":"02:00.165","Text":"This is e to the power of infinity,"},{"Start":"02:00.165 ","End":"02:01.235","Text":"and as we said here,"},{"Start":"02:01.235 ","End":"02:02.960","Text":"this is equal to infinity."},{"Start":"02:02.960 ","End":"02:05.750","Text":"On the other hand, if x goes to 0 from the left,"},{"Start":"02:05.750 ","End":"02:09.830","Text":"then here we get e to the power of 1 over 0 minus,"},{"Start":"02:09.830 ","End":"02:12.415","Text":"which is close to 0 but negative."},{"Start":"02:12.415 ","End":"02:16.880","Text":"That\u0027s 1 over 0 minus is known to be minus infinity."},{"Start":"02:16.880 ","End":"02:20.255","Text":"Here we get e to the minus infinity,"},{"Start":"02:20.255 ","End":"02:22.960","Text":"and as we said here that\u0027s equal to 0."},{"Start":"02:22.960 ","End":"02:25.070","Text":"The thing is we took the right limit and"},{"Start":"02:25.070 ","End":"02:27.995","Text":"the left limit and they turned out to be different."},{"Start":"02:27.995 ","End":"02:30.585","Text":"This and this were not equal,"},{"Start":"02:30.585 ","End":"02:33.560","Text":"and when the left limit does not equal the right limit,"},{"Start":"02:33.560 ","End":"02:40.220","Text":"then there is no limit or limit doesn\u0027t exist or something like that, and that\u0027s it."}],"ID":4818},{"Watched":false,"Name":"Exercise 8","Duration":"1m 19s","ChapterTopicVideoID":4819,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we have to find the limit as x goes to 0 plus,"},{"Start":"00:04.530 ","End":"00:08.190","Text":"which means x goes to 0 from the right of this expression,"},{"Start":"00:08.190 ","End":"00:10.905","Text":"1 over 1 plus 2^1 over x."},{"Start":"00:10.905 ","End":"00:14.115","Text":"This is a simple case of substitution."},{"Start":"00:14.115 ","End":"00:17.280","Text":"We substitute the symbol 0 plus,"},{"Start":"00:17.280 ","End":"00:26.280","Text":"this equals 1 over 1 plus 2^1 over 0 plus."},{"Start":"00:26.280 ","End":"00:33.000","Text":"Now, this equals 1 over 1 plus 2 to the power of,"},{"Start":"00:33.000 ","End":"00:34.545","Text":"now 1 over 0 plus,"},{"Start":"00:34.545 ","End":"00:37.200","Text":"you should remember that this equals plus infinity."},{"Start":"00:37.200 ","End":"00:38.600","Text":"Just remember, if it\u0027s a very,"},{"Start":"00:38.600 ","End":"00:40.380","Text":"very small positive number,"},{"Start":"00:40.380 ","End":"00:42.770","Text":"1 over it, the reciprocal would be a very,"},{"Start":"00:42.770 ","End":"00:44.580","Text":"very large positive number."},{"Start":"00:44.580 ","End":"00:46.575","Text":"In general, this is infinity."},{"Start":"00:46.575 ","End":"00:50.445","Text":"Now, 2^infinity is infinity,"},{"Start":"00:50.445 ","End":"00:56.150","Text":"so that\u0027s equal to 1 over 1 plus infinity."},{"Start":"00:56.150 ","End":"00:57.530","Text":"Continue on the next line,"},{"Start":"00:57.530 ","End":"01:00.230","Text":"and that\u0027s equal to 1 over."},{"Start":"01:00.230 ","End":"01:03.080","Text":"Remember, infinity or minus infinity,"},{"Start":"01:03.080 ","End":"01:07.250","Text":"if you add or subtract a finite quantity, it stays the same,"},{"Start":"01:07.250 ","End":"01:09.665","Text":"so 1 plus infinity is just infinity,"},{"Start":"01:09.665 ","End":"01:14.735","Text":"and 1 over infinity is equal to 0 plus, well, just 0."},{"Start":"01:14.735 ","End":"01:18.035","Text":"1 divided by an enormous quantity gives us 0,"},{"Start":"01:18.035 ","End":"01:20.610","Text":"and 0 is our answer."}],"ID":4819},{"Watched":false,"Name":"Exercise 9","Duration":"1m 25s","ChapterTopicVideoID":4743,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we have to find the limit as x goes to 0 minus,"},{"Start":"00:04.260 ","End":"00:09.915","Text":"which means x goes to 0 from the left limit of 1 over 1 plus 2^1 over x."},{"Start":"00:09.915 ","End":"00:14.895","Text":"We can do this simply by substituting a set of x to 0 minus."},{"Start":"00:14.895 ","End":"00:17.250","Text":"It\u0027s very similar to the previous exercise where"},{"Start":"00:17.250 ","End":"00:19.740","Text":"we had a 0 plus here, but there\u0027s a difference."},{"Start":"00:19.740 ","End":"00:27.570","Text":"This equals 1 over 1 plus 2^1 over 0 minus,"},{"Start":"00:27.570 ","End":"00:34.980","Text":"and this equals 1 over 1 plus 2^1 over 0 minus is minus infinity,"},{"Start":"00:34.980 ","End":"00:37.730","Text":"1 over a very, very small but negative quantity is"},{"Start":"00:37.730 ","End":"00:41.610","Text":"a very large negative quantity and this gives us minus infinity."},{"Start":"00:41.610 ","End":"00:43.650","Text":"Let\u0027s see if we can figure out the side,"},{"Start":"00:43.650 ","End":"00:45.780","Text":"what is 2^minus infinity?"},{"Start":"00:45.780 ","End":"00:50.614","Text":"2^minus infinity is equal to because the rules of exponents,"},{"Start":"00:50.614 ","End":"00:54.140","Text":"1 over 2^infinity and we mentioned before"},{"Start":"00:54.140 ","End":"00:58.670","Text":"that 2^infinity is infinity just like e^infinity."},{"Start":"00:58.670 ","End":"01:06.225","Text":"In fact, what you should know is that a^infinity is infinity if a is bigger than 1."},{"Start":"01:06.225 ","End":"01:07.675","Text":"Any number larger than 1,"},{"Start":"01:07.675 ","End":"01:11.630","Text":"you keep multiplying by itself goes larger and larger and 1 over infinity,"},{"Start":"01:11.630 ","End":"01:13.400","Text":"of course, is 0."},{"Start":"01:13.400 ","End":"01:17.510","Text":"I can put this 2 to the minus infinity equals 0 back here and what"},{"Start":"01:17.510 ","End":"01:22.205","Text":"I\u0027ll get is equal to 1 over 1 plus 0,"},{"Start":"01:22.205 ","End":"01:26.190","Text":"which is just equal to 1. That\u0027s the answer."}],"ID":4752},{"Watched":false,"Name":"Exercise 10","Duration":"1m 5s","ChapterTopicVideoID":4744,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.000 ","End":"00:06.630","Text":"0 of 1 over 1 plus 2 to the 1 over x."},{"Start":"00:06.630 ","End":"00:10.650","Text":"Now I\u0027ve highlighted the 1 over x because this 1 over x fits"},{"Start":"00:10.650 ","End":"00:16.665","Text":"the definition when x goes to 0 of something which is not equal to 0,"},{"Start":"00:16.665 ","End":"00:19.455","Text":"non-zero over something which is 0."},{"Start":"00:19.455 ","End":"00:21.150","Text":"When we see this kind of situation,"},{"Start":"00:21.150 ","End":"00:24.960","Text":"we have to separate the limits into the limit from the left and limit from the right."},{"Start":"00:24.960 ","End":"00:27.540","Text":"In other words, we have to compute 2 limits."},{"Start":"00:27.540 ","End":"00:32.715","Text":"1 is the limit as x goes to 0 from the right,"},{"Start":"00:32.715 ","End":"00:36.940","Text":"and the other is the limit as x goes to 0 from the left."},{"Start":"00:36.940 ","End":"00:42.500","Text":"Now, these 2 limits are exactly the previous 2 exercises and if you haven\u0027t done them,"},{"Start":"00:42.500 ","End":"00:45.080","Text":"I suggest you take a pause and review them"},{"Start":"00:45.080 ","End":"00:47.990","Text":"because this is exactly the exercise before last."},{"Start":"00:47.990 ","End":"00:52.144","Text":"We got the answer here to equal 0 and in this exercise,"},{"Start":"00:52.144 ","End":"00:53.585","Text":"the answer was equal to 1."},{"Start":"00:53.585 ","End":"00:56.990","Text":"The point now is that the right limit and the left limit are not equal."},{"Start":"00:56.990 ","End":"00:58.610","Text":"This and this, if I compare them,"},{"Start":"00:58.610 ","End":"01:00.650","Text":"they are not equal and whenever"},{"Start":"01:00.650 ","End":"01:04.225","Text":"the left limit is not equal to the right limit then there is no limit,"},{"Start":"01:04.225 ","End":"01:06.540","Text":"so there is no limit."}],"ID":4753},{"Watched":false,"Name":"Exercise 11","Duration":"1m 25s","ChapterTopicVideoID":4745,"CourseChapterTopicPlaylistID":3699,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we have to find the limit as x goes to 0 from the right or"},{"Start":"00:04.080 ","End":"00:08.820","Text":"x goes to 0 plus of natural log of x times the cotangent of x."},{"Start":"00:08.820 ","End":"00:13.800","Text":"All we have to do is substitute 0 plus here and here."},{"Start":"00:13.800 ","End":"00:17.495","Text":"What we get is we get the natural log."},{"Start":"00:17.495 ","End":"00:26.630","Text":"This equals the natural log of x 0 plus times the cotangent of 0 plus."},{"Start":"00:26.630 ","End":"00:28.880","Text":"Let\u0027s see. Now this,"},{"Start":"00:28.880 ","End":"00:36.050","Text":"we\u0027ve seen already many times the natural log of 0 plus is minus infinity times."},{"Start":"00:36.050 ","End":"00:39.550","Text":"Let\u0027s do this as an exercise at the side."},{"Start":"00:39.550 ","End":"00:42.560","Text":"You can remember it, but if you don\u0027t,"},{"Start":"00:42.560 ","End":"00:46.745","Text":"we can just write it as cotangent is cosine over sine."},{"Start":"00:46.745 ","End":"00:54.080","Text":"Cosine of 0 plus sine of 0 plus now the cosine of"},{"Start":"00:54.080 ","End":"01:01.915","Text":"0 is 1 and the sine of 0 plus is just 0 plus."},{"Start":"01:01.915 ","End":"01:04.100","Text":"The sine of something very, very small,"},{"Start":"01:04.100 ","End":"01:07.700","Text":"but close to 0 is also very small and positive."},{"Start":"01:07.700 ","End":"01:11.560","Text":"This thing is equal to plus infinity,"},{"Start":"01:11.560 ","End":"01:15.695","Text":"1 over 0 plus is plus infinity and so I can"},{"Start":"01:15.695 ","End":"01:20.015","Text":"write this as minus infinity times plus infinity,"},{"Start":"01:20.015 ","End":"01:26.090","Text":"which gives me minus infinity and that\u0027s the answer."}],"ID":4754}],"Thumbnail":null,"ID":3699},{"Name":"Technique 5 X Tends to Infinity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"X Tends to Infinity Part 1","Duration":"10m 23s","ChapterTopicVideoID":9303,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"In this clip, we talk about technique number 5 for limits."},{"Start":"00:03.705 ","End":"00:07.965","Text":"What to do when x tends to infinity and we write"},{"Start":"00:07.965 ","End":"00:13.110","Text":"this for a function f as limit as x tends to infinity of f of x."},{"Start":"00:13.110 ","End":"00:17.850","Text":"But of course there could also be limit as x tends to minus infinity of f of x."},{"Start":"00:17.850 ","End":"00:20.670","Text":"There is no such number as infinity,"},{"Start":"00:20.670 ","End":"00:23.550","Text":"but infinity in some ways behaves like a number."},{"Start":"00:23.550 ","End":"00:26.400","Text":"You can think of it as a giant, huge,"},{"Start":"00:26.400 ","End":"00:30.660","Text":"enormous number which keeps growing boundlessly,"},{"Start":"00:30.660 ","End":"00:32.490","Text":"no limit to the size."},{"Start":"00:32.490 ","End":"00:34.185","Text":"That\u0027s a rough idea."},{"Start":"00:34.185 ","End":"00:37.255","Text":"Nevertheless, it behaves in many ways like a number and there"},{"Start":"00:37.255 ","End":"00:40.469","Text":"even some rules which I\u0027d like to call the arithmetic of infinity."},{"Start":"00:40.469 ","End":"00:45.125","Text":"So I\u0027m going to write these rules here and call it the arithmetic of infinity."},{"Start":"00:45.125 ","End":"00:47.990","Text":"So let me start with number 1,"},{"Start":"00:47.990 ","End":"00:52.880","Text":"infinity plus infinity equals infinity,"},{"Start":"00:52.880 ","End":"00:56.120","Text":"a number which is huge and keeps growing boundlessly plus"},{"Start":"00:56.120 ","End":"01:00.145","Text":"another 1 of those is still going to be a huge number that grows without end."},{"Start":"01:00.145 ","End":"01:05.210","Text":"Number 2, infinity plus a is also equal to infinity,"},{"Start":"01:05.210 ","End":"01:08.180","Text":"where a is just an actual number something that keeps"},{"Start":"01:08.180 ","End":"01:12.110","Text":"growing endlessly and boundlessly if you add some constant to it,"},{"Start":"01:12.110 ","End":"01:17.135","Text":"like 7 or even 7 billion is still going to keep growing boundlessly."},{"Start":"01:17.135 ","End":"01:21.290","Text":"Number 3, infinity times a."},{"Start":"01:21.290 ","End":"01:22.675","Text":"This is a tricky 1,"},{"Start":"01:22.675 ","End":"01:29.345","Text":"because a could be positive or negative and this is defined piecewise if you like."},{"Start":"01:29.345 ","End":"01:31.475","Text":"So it\u0027s equal to infinity,"},{"Start":"01:31.475 ","End":"01:34.190","Text":"provided that a is a positive number."},{"Start":"01:34.190 ","End":"01:40.640","Text":"But to minus infinity if a is a negative number, what if a equals 0?"},{"Start":"01:40.640 ","End":"01:43.580","Text":"Well, that\u0027s 1 of those undefined cases"},{"Start":"01:43.580 ","End":"01:46.205","Text":"it\u0027s a meaningless we don\u0027t have infinity times 0."},{"Start":"01:46.205 ","End":"01:48.985","Text":"Next, we come to number 4,"},{"Start":"01:48.985 ","End":"01:53.810","Text":"which is infinity times infinity and again, clearly this huge,"},{"Start":"01:53.810 ","End":"01:58.610","Text":"growing boundless times another 1 of those is still going to be huge,"},{"Start":"01:58.610 ","End":"02:01.970","Text":"enormous, and boundless that\u0027s infinity."},{"Start":"02:01.970 ","End":"02:08.435","Text":"Number 5, the square root of infinity is also equal to infinity."},{"Start":"02:08.435 ","End":"02:11.735","Text":"Because if this keeps growing to a million,"},{"Start":"02:11.735 ","End":"02:13.565","Text":"a billion, a trillion,"},{"Start":"02:13.565 ","End":"02:15.920","Text":"the square root of it will grow a bit more slowly,"},{"Start":"02:15.920 ","End":"02:19.245","Text":"but it will still reach any size that you want."},{"Start":"02:19.245 ","End":"02:20.929","Text":"These 5 are basic,"},{"Start":"02:20.929 ","End":"02:24.080","Text":"there are another 3 which are going to include because they\u0027re convenient,"},{"Start":"02:24.080 ","End":"02:26.524","Text":"but they can be derived from these easily."},{"Start":"02:26.524 ","End":"02:35.585","Text":"Another 1 will be that infinity times minus infinity is equal to minus infinity."},{"Start":"02:35.585 ","End":"02:39.080","Text":"This follows because if infinity times infinity is infinity and"},{"Start":"02:39.080 ","End":"02:42.470","Text":"I put my a as minus 1 and then I use this rule,"},{"Start":"02:42.470 ","End":"02:44.194","Text":"I\u0027ll get the minus infinity."},{"Start":"02:44.194 ","End":"02:48.485","Text":"Similarly, minus infinity times minus infinity,"},{"Start":"02:48.485 ","End":"02:52.670","Text":"I can just use this rule here twice with a equals minus 1."},{"Start":"02:52.670 ","End":"02:55.115","Text":"So this time it will equal plus infinity so"},{"Start":"02:55.115 ","End":"02:58.500","Text":"the plus minuses behaved like with actual numbers."},{"Start":"02:58.500 ","End":"03:00.830","Text":"Finally, to be consistent that we have"},{"Start":"03:00.830 ","End":"03:04.490","Text":"a square root formula that\u0027s also have an infinity squared"},{"Start":"03:04.490 ","End":"03:11.240","Text":"formula and this is equal to infinity and actually this is just a rephrasing of this 1."},{"Start":"03:11.240 ","End":"03:13.565","Text":"So these are the rules of infinity."},{"Start":"03:13.565 ","End":"03:17.000","Text":"There is another 1 I forgot, number 9,"},{"Start":"03:17.000 ","End":"03:21.200","Text":"that a divided by plus or"},{"Start":"03:21.200 ","End":"03:28.550","Text":"minus infinity is equal to 0 and a can be any number plus minus or 0."},{"Start":"03:28.550 ","End":"03:31.820","Text":"There are also some undefined expressions"},{"Start":"03:31.820 ","End":"03:34.520","Text":"involving infinity and I\u0027ll give you some examples,"},{"Start":"03:34.520 ","End":"03:38.480","Text":"I\u0027ll just stress that these are undefined."},{"Start":"03:38.480 ","End":"03:45.515","Text":"For example, infinity minus infinity, 2 large, huge,"},{"Start":"03:45.515 ","End":"03:49.130","Text":"growing unbounded quantities fighting against each other,"},{"Start":"03:49.130 ","End":"03:51.290","Text":"1 pulling upwards, 1 pulling downwards,"},{"Start":"03:51.290 ","End":"03:54.410","Text":"not defined and I\u0027ll just list the rest,"},{"Start":"03:54.410 ","End":"03:58.400","Text":"I won\u0027t go into explanation, 1^infinity is undefined."},{"Start":"03:58.400 ","End":"04:02.240","Text":"Infinity over infinity is undefined,"},{"Start":"04:02.240 ","End":"04:07.190","Text":"and infinity^0 is undefined and maybe there\u0027s some more I forgot."},{"Start":"04:07.190 ","End":"04:11.750","Text":"Let\u0027s get back to the original problem that we were here to solve,"},{"Start":"04:11.750 ","End":"04:14.345","Text":"which was the limit of this form."},{"Start":"04:14.345 ","End":"04:17.570","Text":"But there\u0027s no single technique that will work for all functions"},{"Start":"04:17.570 ","End":"04:21.305","Text":"f and we\u0027re going to concentrate on 1 very common kind of f,"},{"Start":"04:21.305 ","End":"04:22.700","Text":"which is very suitable,"},{"Start":"04:22.700 ","End":"04:29.460","Text":"amenable to this technique and that is when f of x is a polynomial over a polynomial."},{"Start":"04:29.460 ","End":"04:32.450","Text":"But in case you don\u0027t know what a polynomial is, I\u0027ll spell it out."},{"Start":"04:32.450 ","End":"04:36.965","Text":"The numerator might be a times x^n."},{"Start":"04:36.965 ","End":"04:42.240","Text":"Think of it as a generalization of the quadratic ax squared plus bx plus c only"},{"Start":"04:42.240 ","End":"04:49.090","Text":"generalize ax^n plus bx^n minus 1 plus,"},{"Start":"04:49.090 ","End":"04:54.080","Text":"if there is another term may be cx^n minus 2, and so on."},{"Start":"04:54.080 ","End":"04:56.210","Text":"Until we get down to the constant."},{"Start":"04:56.210 ","End":"04:59.060","Text":"I don\u0027t know what letter of the alphabet it is so"},{"Start":"04:59.060 ","End":"05:02.195","Text":"let\u0027s just call it square box or something,"},{"Start":"05:02.195 ","End":"05:06.215","Text":"and same in the denominator also a polynomial,"},{"Start":"05:06.215 ","End":"05:07.865","Text":"let\u0027s use capital letters."},{"Start":"05:07.865 ","End":"05:14.450","Text":"A and it starts my different letter may be x^m plus"},{"Start":"05:14.450 ","End":"05:21.350","Text":"bx^m minus 1 another constant times the next power down, m minus 2."},{"Start":"05:21.350 ","End":"05:24.380","Text":"Again, I don\u0027t know how big this is and where it\u0027s going to end,"},{"Start":"05:24.380 ","End":"05:26.420","Text":"let\u0027s call it just a triangle or something."},{"Start":"05:26.420 ","End":"05:32.840","Text":"So this is called a rational function or a polynomial over a polynomial."},{"Start":"05:32.840 ","End":"05:37.650","Text":"But don\u0027t worry about the names and also if the notation confuses you,"},{"Start":"05:37.650 ","End":"05:40.250","Text":"next example will clarify all that."},{"Start":"05:40.250 ","End":"05:42.425","Text":"So let\u0027s take as an example,"},{"Start":"05:42.425 ","End":"05:50.670","Text":"the limit as x goes to infinity of 4x squared plus 10x plus 1,"},{"Start":"05:50.670 ","End":"05:54.425","Text":"there\u0027s a nice quadratic polynomial over another 1,"},{"Start":"05:54.425 ","End":"05:59.720","Text":"2x squared minus 5x plus 100."},{"Start":"05:59.720 ","End":"06:01.955","Text":"It turns out that all of these,"},{"Start":"06:01.955 ","End":"06:05.660","Text":"what I called rational function or polynomial over polynomial,"},{"Start":"06:05.660 ","End":"06:10.235","Text":"and they all fall into the category of the infinity over infinity."},{"Start":"06:10.235 ","End":"06:15.274","Text":"Now whenever we have an f of x as what I call the rational function,"},{"Start":"06:15.274 ","End":"06:18.605","Text":"which looks like this and in particular this,"},{"Start":"06:18.605 ","End":"06:22.820","Text":"there is a specific technique we use so I\u0027ll just write this out."},{"Start":"06:22.820 ","End":"06:28.040","Text":"That for this kind of exercise the technique is to take the highest power"},{"Start":"06:28.040 ","End":"06:33.320","Text":"of x outside the brackets and I\u0027ll soon explain what this means."},{"Start":"06:33.320 ","End":"06:36.965","Text":"In fact, I\u0027ll illustrate it by means of this particular exercise."},{"Start":"06:36.965 ","End":"06:40.940","Text":"The highest power of x here is x squared because the rest"},{"Start":"06:40.940 ","End":"06:44.990","Text":"is just x and this is a constant and here the highest power is x squared."},{"Start":"06:44.990 ","End":"06:48.830","Text":"So separately on the top and on the bottom we take the highest power,"},{"Start":"06:48.830 ","End":"06:52.895","Text":"which just happens to be the same in both cases, it\u0027s x squared."},{"Start":"06:52.895 ","End":"06:56.525","Text":"So this thing continuing down here,"},{"Start":"06:56.525 ","End":"07:03.350","Text":"and this is equal to the limit as x goes to infinity of x"},{"Start":"07:03.350 ","End":"07:07.280","Text":"squared and we\u0027ll take the rest outside the brackets and"},{"Start":"07:07.280 ","End":"07:11.915","Text":"likewise on the bottom we\u0027ll have x squared times something in brackets."},{"Start":"07:11.915 ","End":"07:13.760","Text":"In this case, the top,"},{"Start":"07:13.760 ","End":"07:16.265","Text":"take x squared out, you\u0027re left with 4."},{"Start":"07:16.265 ","End":"07:18.050","Text":"Take x squared out of here,"},{"Start":"07:18.050 ","End":"07:20.000","Text":"you\u0027re left with 10 over x."},{"Start":"07:20.000 ","End":"07:23.615","Text":"Here, we\u0027re left with 1 over x squared."},{"Start":"07:23.615 ","End":"07:26.000","Text":"On the bottom, take x squared out,"},{"Start":"07:26.000 ","End":"07:36.035","Text":"we\u0027re left with 2 minus 5 over x plus 100 over x squared close the brackets."},{"Start":"07:36.035 ","End":"07:37.775","Text":"Now how does this help us?"},{"Start":"07:37.775 ","End":"07:39.035","Text":"Well, this helps us."},{"Start":"07:39.035 ","End":"07:43.730","Text":"First of all, we can proceed by canceling"},{"Start":"07:43.730 ","End":"07:49.100","Text":"x squared with x squared because x is not 0 its far from 0,"},{"Start":"07:49.100 ","End":"07:52.400","Text":"it\u0027s on its way to infinity and then here everywhere else we can"},{"Start":"07:52.400 ","End":"07:56.870","Text":"substitute and using the rules that we had for the arithmetic of infinity."},{"Start":"07:56.870 ","End":"08:00.830","Text":"This comes out to be on the top we get 4,"},{"Start":"08:00.830 ","End":"08:05.120","Text":"10 over x is a over infinity that was the rule and that was"},{"Start":"08:05.120 ","End":"08:10.765","Text":"0 and also x squared is infinity and 1 over infinity is likewise 0."},{"Start":"08:10.765 ","End":"08:12.800","Text":"Following the same logic on the bottom,"},{"Start":"08:12.800 ","End":"08:16.625","Text":"we get 2 minus 0 plus 0."},{"Start":"08:16.625 ","End":"08:19.470","Text":"So altogether it\u0027s 4 over 2,"},{"Start":"08:19.470 ","End":"08:24.365","Text":"which is equal to 2 and that\u0027s the answer to this particular exercise."},{"Start":"08:24.365 ","End":"08:30.455","Text":"In general, what we do is we take the highest power of x,"},{"Start":"08:30.455 ","End":"08:37.430","Text":"which in this case would be the x^n and the highest power of x here is x^m and take"},{"Start":"08:37.430 ","End":"08:40.280","Text":"this outside the brackets and get"},{"Start":"08:40.280 ","End":"08:44.645","Text":"this thing times something in brackets and then everything else follows from there."},{"Start":"08:44.645 ","End":"08:46.700","Text":"I\u0027d like to clarify something."},{"Start":"08:46.700 ","End":"08:48.515","Text":"When I wrote what I wrote here,"},{"Start":"08:48.515 ","End":"08:51.380","Text":"I was referring to the polynomial over"},{"Start":"08:51.380 ","End":"08:56.420","Text":"polynomial case and that\u0027s what I meant by this kind of exercise."},{"Start":"08:56.420 ","End":"09:02.165","Text":"Here we take the highest power outside the brackets and usually works."},{"Start":"09:02.165 ","End":"09:05.750","Text":"But it\u0027s not restricted to just polynomial over polynomial."},{"Start":"09:05.750 ","End":"09:09.440","Text":"There are other examples where involving"},{"Start":"09:09.440 ","End":"09:14.720","Text":"polynomials where you can take the highest power of x outside the brackets and it helps."},{"Start":"09:14.720 ","End":"09:19.000","Text":"So I\u0027d just like to add an extra little clause here."},{"Start":"09:19.000 ","End":"09:21.920","Text":"Here\u0027s the extra clothes for this kind of exercise,"},{"Start":"09:21.920 ","End":"09:24.995","Text":"as well as others involving polynomials."},{"Start":"09:24.995 ","End":"09:30.740","Text":"But it does work best for the case polynomial over polynomial and in fact,"},{"Start":"09:30.740 ","End":"09:33.080","Text":"in these cases is even a shortcut,"},{"Start":"09:33.080 ","End":"09:34.925","Text":"I\u0027ll show you in a second."},{"Start":"09:34.925 ","End":"09:39.320","Text":"Notice where this 2 came from in this particular example."},{"Start":"09:39.320 ","End":"09:40.835","Text":"If we follow it back,"},{"Start":"09:40.835 ","End":"09:42.755","Text":"it came from 4 over 2,"},{"Start":"09:42.755 ","End":"09:44.870","Text":"which came from here,"},{"Start":"09:44.870 ","End":"09:50.330","Text":"which ultimately came from 4x squared over 2x squared and this is a general rule that"},{"Start":"09:50.330 ","End":"09:55.955","Text":"when you have a polynomial over polynomial and you take out just the highest,"},{"Start":"09:55.955 ","End":"09:58.910","Text":"in this case, the highest power is 4x squared."},{"Start":"09:58.910 ","End":"10:01.190","Text":"In this case, the highest power is"},{"Start":"10:01.190 ","End":"10:04.580","Text":"2x squared be careful though it needn\u0027t be the first term."},{"Start":"10:04.580 ","End":"10:07.250","Text":"The highest power could be somewhere in the middle and if we just"},{"Start":"10:07.250 ","End":"10:10.759","Text":"compute the limit of these highest powers,"},{"Start":"10:10.759 ","End":"10:13.565","Text":"1 from the top, 1 from the bottom and ignore all the rest,"},{"Start":"10:13.565 ","End":"10:15.140","Text":"then you\u0027ll get the same answer."},{"Start":"10:15.140 ","End":"10:17.660","Text":"So that\u0027s something very useful,"},{"Start":"10:17.660 ","End":"10:21.740","Text":"very practical and soon we\u0027ll see this in the exercises."},{"Start":"10:21.740 ","End":"10:24.360","Text":"We\u0027ll have some examples."}],"ID":9615},{"Watched":false,"Name":"X Tends to Infinity Part 2","Duration":"7m 40s","ChapterTopicVideoID":9304,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"Let\u0027s proceed with the examples."},{"Start":"00:02.490 ","End":"00:05.700","Text":"All of these examples on this page will be polynomials"},{"Start":"00:05.700 ","End":"00:08.970","Text":"over polynomials and we\u0027ll practice that shortcut."},{"Start":"00:08.970 ","End":"00:14.370","Text":"The first 1 is as written here and the first thing we do is to find"},{"Start":"00:14.370 ","End":"00:20.800","Text":"the highest power of x in each and very quickly we find that it\u0027s this 1, the x^4."},{"Start":"00:20.800 ","End":"00:24.765","Text":"We circled the whole term including the coefficient."},{"Start":"00:24.765 ","End":"00:32.160","Text":"On the bottom, we find that it\u0027s the 100x^4, we circle that."},{"Start":"00:32.160 ","End":"00:34.745","Text":"Then we continue to do the limit,"},{"Start":"00:34.745 ","End":"00:36.635","Text":"ignoring all the other terms."},{"Start":"00:36.635 ","End":"00:41.540","Text":"This is the limit as x goes to infinity of"},{"Start":"00:41.540 ","End":"00:48.060","Text":"10x^4, over 100x^4."},{"Start":"00:48.060 ","End":"00:52.640","Text":"Coincidentally, again we have an equal power at the top and the bottom,"},{"Start":"00:52.640 ","End":"00:56.180","Text":"x^4 and x^4, and the whole thing cancels and we\u0027re left with"},{"Start":"00:56.180 ","End":"01:00.085","Text":"just 10/100, which is 1/10."},{"Start":"01:00.085 ","End":"01:02.040","Text":"Now, the next example."},{"Start":"01:02.040 ","End":"01:09.210","Text":"Limit as x goes to infinity of 2x squared minus 5 over 3x plus 7."},{"Start":"01:09.210 ","End":"01:11.565","Text":"Circle the highest powers,"},{"Start":"01:11.565 ","End":"01:13.680","Text":"in this case, 2x squared,"},{"Start":"01:13.680 ","End":"01:16.290","Text":"in this case the 3x."},{"Start":"01:16.290 ","End":"01:19.695","Text":"Then this is equal to limit,"},{"Start":"01:19.695 ","End":"01:22.315","Text":"and we just throw out the rest of the terms,"},{"Start":"01:22.315 ","End":"01:27.600","Text":"of 2x squared over 3x."},{"Start":"01:27.600 ","End":"01:30.940","Text":"Now we do have different powers at the top and the bottom,"},{"Start":"01:30.940 ","End":"01:32.255","Text":"the top is higher."},{"Start":"01:32.255 ","End":"01:34.460","Text":"Still we can do a partial cancellation,"},{"Start":"01:34.460 ","End":"01:35.975","Text":"x squared over x,"},{"Start":"01:35.975 ","End":"01:39.715","Text":"this gives us just the x on the top."},{"Start":"01:39.715 ","End":"01:46.115","Text":"Then I substitute x equals infinity and we get 2/3 of infinity basically."},{"Start":"01:46.115 ","End":"01:53.095","Text":"We do the arithmetic for infinity and positive number times infinity is infinity."},{"Start":"01:53.095 ","End":"01:56.845","Text":"That\u0027s this 1. Now on to the next 1."},{"Start":"01:56.845 ","End":"02:00.050","Text":"Limit as x goes to minus infinity,"},{"Start":"02:00.050 ","End":"02:03.680","Text":"this time of 2x squared plus 1 over 10x."},{"Start":"02:03.680 ","End":"02:07.350","Text":"We circle the highest powers,"},{"Start":"02:07.350 ","End":"02:09.135","Text":"this is the usual technique."},{"Start":"02:09.135 ","End":"02:11.780","Text":"Here it\u0027s this and here there is only 1 term,"},{"Start":"02:11.780 ","End":"02:13.675","Text":"so must be this."},{"Start":"02:13.675 ","End":"02:16.395","Text":"Then we continue, limit,"},{"Start":"02:16.395 ","End":"02:18.300","Text":"still x goes to infinity,"},{"Start":"02:18.300 ","End":"02:23.025","Text":"of 2x squared over 10x."},{"Start":"02:23.025 ","End":"02:27.170","Text":"Once again, we have a higher power in the top,"},{"Start":"02:27.170 ","End":"02:29.280","Text":"but we can partially cancel,"},{"Start":"02:29.280 ","End":"02:31.515","Text":"we even got x squared and x again."},{"Start":"02:31.515 ","End":"02:34.020","Text":"This x cancels 1 of the x\u0027s here."},{"Start":"02:34.020 ","End":"02:37.710","Text":"It\u0027s like I break through the 2 and this equals,"},{"Start":"02:37.710 ","End":"02:39.585","Text":"if I put x equals infinity,"},{"Start":"02:39.585 ","End":"02:43.970","Text":"2/10 times infinity and this is on these rules"},{"Start":"02:43.970 ","End":"02:48.470","Text":"that a positive number times infinity is equal to infinity,"},{"Start":"02:48.470 ","End":"02:50.755","Text":"and that\u0027s this 1."},{"Start":"02:50.755 ","End":"02:52.480","Text":"Let\u0027s do another 1."},{"Start":"02:52.480 ","End":"02:56.450","Text":"Limit as x goes to infinity of all this stuff."},{"Start":"02:56.450 ","End":"02:58.460","Text":"Identify the highest powers,"},{"Start":"02:58.460 ","End":"03:00.725","Text":"they are the only ones that matter."},{"Start":"03:00.725 ","End":"03:04.490","Text":"In this case, it\u0027s the x^4 term,"},{"Start":"03:04.490 ","End":"03:07.165","Text":"the whole term I circle."},{"Start":"03:07.165 ","End":"03:09.510","Text":"Here that\u0027s a 5 there,"},{"Start":"03:09.510 ","End":"03:13.110","Text":"this is x^5, that\u0027s obviously the highest term, circle this."},{"Start":"03:13.110 ","End":"03:17.950","Text":"Then just write it as the limit x goes to"},{"Start":"03:17.950 ","End":"03:24.700","Text":"infinity of 2x^4 over 10x^5."},{"Start":"03:24.700 ","End":"03:28.705","Text":"In this case, the bottom has the higher power this time,"},{"Start":"03:28.705 ","End":"03:31.120","Text":"this is x^5, this is x^4."},{"Start":"03:31.120 ","End":"03:33.670","Text":"When we cancel x^4,"},{"Start":"03:33.670 ","End":"03:36.250","Text":"cancels 2x^5 x times."},{"Start":"03:36.250 ","End":"03:38.985","Text":"I can just cross out the 5."},{"Start":"03:38.985 ","End":"03:43.065","Text":"What I get is 1/5 over x,"},{"Start":"03:43.065 ","End":"03:45.960","Text":"the x is infinity like that."},{"Start":"03:45.960 ","End":"03:52.905","Text":"a over infinity for any a is equal to 0. That\u0027s this 1."},{"Start":"03:52.905 ","End":"03:56.405","Text":"Now I\u0027d like to and I\u0027ll just do it verbally I won\u0027t write it down,"},{"Start":"03:56.405 ","End":"04:01.845","Text":"we\u0027ve used the ultimate shortcut for when you have a polynomial over a polynomial."},{"Start":"04:01.845 ","End":"04:07.880","Text":"This is the rule, if the highest powers are equal x^4 and x^4,"},{"Start":"04:07.880 ","End":"04:11.665","Text":"then the answer is just the 10/100,"},{"Start":"04:11.665 ","End":"04:13.395","Text":"just take the coefficients,"},{"Start":"04:13.395 ","End":"04:16.320","Text":"10/ 100, which is 1/10."},{"Start":"04:16.320 ","End":"04:19.730","Text":"If the highest power on"},{"Start":"04:19.730 ","End":"04:24.335","Text":"the numerator is higher than the highest power on the denominator,"},{"Start":"04:24.335 ","End":"04:27.890","Text":"then the answer is going to be plus or minus infinity according to"},{"Start":"04:27.890 ","End":"04:31.860","Text":"the signs and according to whether it\u0027s x goes to plus or minus infinity,"},{"Start":"04:31.860 ","End":"04:34.675","Text":"but other than the plus or minus is going to be infinity."},{"Start":"04:34.675 ","End":"04:36.080","Text":"We had an example here,"},{"Start":"04:36.080 ","End":"04:37.640","Text":"and we had an example here."},{"Start":"04:37.640 ","End":"04:41.029","Text":"The third case is where it\u0027s higher on the bottom,"},{"Start":"04:41.029 ","End":"04:45.980","Text":"like 4 versus 5 and in that case the answer is always going to be 0."},{"Start":"04:45.980 ","End":"04:48.425","Text":"That\u0027s basically your rules."},{"Start":"04:48.425 ","End":"04:50.660","Text":"Now, I\u0027d like to go on to do"},{"Start":"04:50.660 ","End":"04:54.050","Text":"something a little bit different, slightly different example."},{"Start":"04:54.050 ","End":"04:58.295","Text":"I want to break out of this mold of polynomial over polynomial,"},{"Start":"04:58.295 ","End":"05:00.920","Text":"also known as rational function."},{"Start":"05:00.920 ","End":"05:07.500","Text":"I want to do something that isn\u0027t just like that and my example is going to be here."},{"Start":"05:07.500 ","End":"05:10.310","Text":"A different example, not the usual."},{"Start":"05:10.310 ","End":"05:14.120","Text":"Notice, this is a polynomial and this is a polynomial,"},{"Start":"05:14.120 ","End":"05:18.995","Text":"but this 1 is under a square root sign, something completely different."},{"Start":"05:18.995 ","End":"05:20.720","Text":"Now, we can\u0027t use"},{"Start":"05:20.720 ","End":"05:25.820","Text":"that shortcut trick of just taking the highest power in each polynomial,"},{"Start":"05:25.820 ","End":"05:31.930","Text":"but we still can take out the highest power as a factor that\u0027s still holds."},{"Start":"05:31.930 ","End":"05:33.900","Text":"Let\u0027s do that. Now,"},{"Start":"05:33.900 ","End":"05:37.490","Text":"the highest power here is the x squared,"},{"Start":"05:37.490 ","End":"05:40.085","Text":"and the highest power here is the x."},{"Start":"05:40.085 ","End":"05:45.785","Text":"In each case, we take the highest power outside of its corresponding polynomial."},{"Start":"05:45.785 ","End":"05:47.300","Text":"In the first case,"},{"Start":"05:47.300 ","End":"05:48.895","Text":"we\u0027ll get the limit,"},{"Start":"05:48.895 ","End":"05:53.330","Text":"x goes to infinity of the square root."},{"Start":"05:53.330 ","End":"05:56.645","Text":"Now taking x squared outside the brackets,"},{"Start":"05:56.645 ","End":"06:00.645","Text":"what we\u0027re left with is x squared out of x squared leaves 1,"},{"Start":"06:00.645 ","End":"06:04.125","Text":"out of the x leaves 1 over x,"},{"Start":"06:04.125 ","End":"06:07.085","Text":"and here, 1 over x squared."},{"Start":"06:07.085 ","End":"06:14.345","Text":"On the bottom, the x comes out and we\u0027re left with 4 plus 1 over x."},{"Start":"06:14.345 ","End":"06:21.020","Text":"Now, I can take this square root and apply it separately to each factor."},{"Start":"06:21.020 ","End":"06:22.640","Text":"I get the limit,"},{"Start":"06:22.640 ","End":"06:24.710","Text":"still x goes to infinity,"},{"Start":"06:24.710 ","End":"06:27.890","Text":"of the square root of x squared times"},{"Start":"06:27.890 ","End":"06:33.170","Text":"the square root of 1 plus 1 over x plus 1 over x squared,"},{"Start":"06:33.170 ","End":"06:36.400","Text":"and all this is over x."},{"Start":"06:36.400 ","End":"06:40.095","Text":"Here, 4 plus 1 over x."},{"Start":"06:40.095 ","End":"06:42.725","Text":"Now, what\u0027s the square root of x squared?"},{"Start":"06:42.725 ","End":"06:46.775","Text":"Everyone will say x, and it\u0027s true in this case,"},{"Start":"06:46.775 ","End":"06:48.410","Text":"but it\u0027s not always true."},{"Start":"06:48.410 ","End":"06:50.720","Text":"In general just remember that the square root of x"},{"Start":"06:50.720 ","End":"06:53.450","Text":"squared is equal to the absolute value of x."},{"Start":"06:53.450 ","End":"06:56.965","Text":"But of course here x is positive on its way to infinity,"},{"Start":"06:56.965 ","End":"06:59.235","Text":"so it\u0027s certainly positive."},{"Start":"06:59.235 ","End":"07:01.310","Text":"It makes sense that this thing is x,"},{"Start":"07:01.310 ","End":"07:05.900","Text":"in which case this thing cancels and all we\u0027re left with"},{"Start":"07:05.900 ","End":"07:11.700","Text":"is the square root of 1 plus 0 plus 0,"},{"Start":"07:11.700 ","End":"07:16.100","Text":"because all these thing is1 over infinity and 1 over infinity is 0."},{"Start":"07:16.100 ","End":"07:19.460","Text":"Also we\u0027re here 1 over infinity is 0,"},{"Start":"07:19.460 ","End":"07:25.140","Text":"so over 4 and that just leaves us with 1 quarter."},{"Start":"07:25.140 ","End":"07:30.020","Text":"This is the example that\u0027s different from the rest and"},{"Start":"07:30.020 ","End":"07:32.705","Text":"basically we\u0027re done for this clip"},{"Start":"07:32.705 ","End":"07:35.690","Text":"but there are lots of exercises and you need to practice,"},{"Start":"07:35.690 ","End":"07:40.810","Text":"so I\u0027ll leave you to get on with those. That\u0027s all."}],"ID":9616},{"Watched":false,"Name":"Exercise 1","Duration":"1m 29s","ChapterTopicVideoID":4747,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.535","Text":"In this exercise, we have to find the limit as x goes to infinity of e to the minus x,"},{"Start":"00:05.535 ","End":"00:08.310","Text":"all of these to the power of natural log of x."},{"Start":"00:08.310 ","End":"00:13.545","Text":"The usual thing to do with infinity is just to substitute it and if we substituted it,"},{"Start":"00:13.545 ","End":"00:19.455","Text":"what we would get would be e to the power of minus infinity,"},{"Start":"00:19.455 ","End":"00:24.365","Text":"to the power of natural log of infinity."},{"Start":"00:24.365 ","End":"00:27.770","Text":"Now, this equals e to the minus infinity is"},{"Start":"00:27.770 ","End":"00:31.490","Text":"0 and the natural log of infinity is infinity."},{"Start":"00:31.490 ","End":"00:33.290","Text":"We get 0 to the infinity,"},{"Start":"00:33.290 ","End":"00:35.600","Text":"which is another 1 of those indeterminate forms."},{"Start":"00:35.600 ","End":"00:36.680","Text":"We can\u0027t say what it is."},{"Start":"00:36.680 ","End":"00:39.905","Text":"It could be different things in different exercises."},{"Start":"00:39.905 ","End":"00:42.245","Text":"What we\u0027re going do is a tiny trick."},{"Start":"00:42.245 ","End":"00:44.450","Text":"If you remember the rules of exponents,"},{"Start":"00:44.450 ","End":"00:48.170","Text":"what we could do is to write this as the limit,"},{"Start":"00:48.170 ","End":"00:50.175","Text":"x goes to infinity."},{"Start":"00:50.175 ","End":"00:52.190","Text":"When we have an exponent of an exponent,"},{"Start":"00:52.190 ","End":"00:53.660","Text":"you multiply the exponents,"},{"Start":"00:53.660 ","End":"01:00.259","Text":"so we get e to the power of minus x times natural log of x."},{"Start":"01:00.259 ","End":"01:02.240","Text":"Now we do the substitution,"},{"Start":"01:02.240 ","End":"01:04.130","Text":"we get, if x is infinity,"},{"Start":"01:04.130 ","End":"01:07.835","Text":"we get e to the power of minus infinity,"},{"Start":"01:07.835 ","End":"01:09.950","Text":"and the natural log of infinity,"},{"Start":"01:09.950 ","End":"01:11.270","Text":"we\u0027ve already mentioned,"},{"Start":"01:11.270 ","End":"01:13.790","Text":"is infinity times infinity."},{"Start":"01:13.790 ","End":"01:18.440","Text":"Now infinity times infinity is infinity so we get e"},{"Start":"01:18.440 ","End":"01:23.105","Text":"to the minus infinity and e to the power of minus infinity."},{"Start":"01:23.105 ","End":"01:24.955","Text":"We\u0027ve seen that many times."},{"Start":"01:24.955 ","End":"01:29.470","Text":"that\u0027s equal to 0 and that\u0027s our answer."}],"ID":4756},{"Watched":false,"Name":"Exercise 2","Duration":"1m 33s","ChapterTopicVideoID":4748,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.480 ","End":"00:07.830","Text":"minus infinity of arctangent x plus e^x."},{"Start":"00:07.830 ","End":"00:09.375","Text":"In the case of limits,"},{"Start":"00:09.375 ","End":"00:12.405","Text":"if a plus or minus infinity, we just substitute."},{"Start":"00:12.405 ","End":"00:18.270","Text":"What we get is that this thing is equal to the arctangent"},{"Start":"00:18.270 ","End":"00:24.795","Text":"of minus infinity plus e to the power of minus infinity."},{"Start":"00:24.795 ","End":"00:28.920","Text":"The arctangent of minus infinity is known from trigonometry."},{"Start":"00:28.920 ","End":"00:35.310","Text":"From before you\u0027ve seen this is minus Pi over 2 and e to the minus infinity."},{"Start":"00:35.310 ","End":"00:38.610","Text":"We\u0027ve also seen before that this is equal to"},{"Start":"00:38.610 ","End":"00:43.280","Text":"0 and the answer is minus Pi over 2, and we\u0027re done."},{"Start":"00:43.280 ","End":"00:45.620","Text":"But for those who want a bit of further explanation,"},{"Start":"00:45.620 ","End":"00:52.775","Text":"the arctangent function is something like an asymptote at plus and minus Pi over 2."},{"Start":"00:52.775 ","End":"00:55.220","Text":"This is Pi over 2,"},{"Start":"00:55.220 ","End":"00:58.920","Text":"this is minus Pi over 2."},{"Start":"00:58.920 ","End":"01:02.760","Text":"The function basically looks like this."},{"Start":"01:02.760 ","End":"01:05.405","Text":"When something goes to minus infinity,"},{"Start":"01:05.405 ","End":"01:08.675","Text":"the arctangent goes to minus Pi over 2."},{"Start":"01:08.675 ","End":"01:10.130","Text":"As for this, again,"},{"Start":"01:10.130 ","End":"01:16.130","Text":"we can look at the graph and e^x goes something like goes through 0,"},{"Start":"01:16.130 ","End":"01:18.475","Text":"1, something like this."},{"Start":"01:18.475 ","End":"01:21.380","Text":"Here it goes to 0, the y,"},{"Start":"01:21.380 ","End":"01:23.720","Text":"and here it goes to infinity."},{"Start":"01:23.720 ","End":"01:26.720","Text":"It\u0027s like here it goes to Pi over 2,"},{"Start":"01:26.720 ","End":"01:29.690","Text":"and here it goes to minus Pi over 2."},{"Start":"01:29.690 ","End":"01:34.440","Text":"So quick look at the graphs will tell you. We\u0027re done."}],"ID":4757},{"Watched":false,"Name":"Exercise 3","Duration":"2m 9s","ChapterTopicVideoID":4749,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression,"},{"Start":"00:05.280 ","End":"00:07.530","Text":"which is a polynomial over a polynomial."},{"Start":"00:07.530 ","End":"00:08.640","Text":"When x goes to infinity,"},{"Start":"00:08.640 ","End":"00:11.870","Text":"the usual thing to do would be to substitute x equals infinity,"},{"Start":"00:11.870 ","End":"00:13.070","Text":"and if we did that,"},{"Start":"00:13.070 ","End":"00:17.940","Text":"we would get 4 times infinity squared plus 2 over"},{"Start":"00:17.940 ","End":"00:23.160","Text":"infinity squared plus 1,000 times infinity."},{"Start":"00:23.160 ","End":"00:24.540","Text":"If you computed this,"},{"Start":"00:24.540 ","End":"00:27.675","Text":"you\u0027d see that what we get is infinity over infinity,"},{"Start":"00:27.675 ","End":"00:29.220","Text":"which is an indeterminate form."},{"Start":"00:29.220 ","End":"00:30.690","Text":"We can\u0027t say what this is."},{"Start":"00:30.690 ","End":"00:33.330","Text":"So we\u0027re going to have to try another approach."},{"Start":"00:33.330 ","End":"00:35.670","Text":"Let me just erase this."},{"Start":"00:35.670 ","End":"00:38.385","Text":"This is not the way to do it."},{"Start":"00:38.385 ","End":"00:41.000","Text":"Instead, we\u0027re going to use a trick which is often"},{"Start":"00:41.000 ","End":"00:43.580","Text":"used when we have a polynomial over a polynomial,"},{"Start":"00:43.580 ","End":"00:45.320","Text":"and that trick is to take out"},{"Start":"00:45.320 ","End":"00:49.160","Text":"the highest power of x in the numerator and in the denominator."},{"Start":"00:49.160 ","End":"00:53.060","Text":"Rewriting this, we get the limit x"},{"Start":"00:53.060 ","End":"00:57.710","Text":"goes to infinity and the highest power of x here is x squared."},{"Start":"00:57.710 ","End":"00:59.360","Text":"We\u0027ll take that out,"},{"Start":"00:59.360 ","End":"01:00.905","Text":"and that\u0027s x squared."},{"Start":"01:00.905 ","End":"01:04.865","Text":"What we\u0027re left with is here we get a 4 and a 2."},{"Start":"01:04.865 ","End":"01:07.430","Text":"We have to divide by x squared,"},{"Start":"01:07.430 ","End":"01:09.485","Text":"which is what we took out the brackets."},{"Start":"01:09.485 ","End":"01:11.045","Text":"In the denominator,"},{"Start":"01:11.045 ","End":"01:13.055","Text":"the highest power is also x squared,"},{"Start":"01:13.055 ","End":"01:14.360","Text":"so it\u0027s x squared."},{"Start":"01:14.360 ","End":"01:16.310","Text":"This time, we\u0027re left with 1."},{"Start":"01:16.310 ","End":"01:19.265","Text":"If you do a bit of algebra, x over x squared is 1 over x."},{"Start":"01:19.265 ","End":"01:23.539","Text":"So we have here 1,000 over x."},{"Start":"01:23.539 ","End":"01:26.675","Text":"Now, we\u0027re lucky, this thing cancels."},{"Start":"01:26.675 ","End":"01:32.840","Text":"So what we\u0027re left with is 4 plus 2 over infinity"},{"Start":"01:32.840 ","End":"01:39.530","Text":"squared all over 1 plus 1,000 over infinity."},{"Start":"01:39.530 ","End":"01:43.250","Text":"At this point, I just like to remind you of something that in general,"},{"Start":"01:43.250 ","End":"01:44.300","Text":"if we have any number,"},{"Start":"01:44.300 ","End":"01:46.430","Text":"a finite number, not usual number,"},{"Start":"01:46.430 ","End":"01:48.535","Text":"and we divide it by infinity,"},{"Start":"01:48.535 ","End":"01:50.720","Text":"actually, it could be minus infinity."},{"Start":"01:50.720 ","End":"01:54.470","Text":"So I\u0027ll write plus or minus infinity. This equals 0."},{"Start":"01:54.470 ","End":"01:56.540","Text":"Using that fact here,"},{"Start":"01:56.540 ","End":"01:59.285","Text":"what we get is 4 plus,"},{"Start":"01:59.285 ","End":"02:00.800","Text":"and this thing is 0,"},{"Start":"02:00.800 ","End":"02:03.640","Text":"over 1 plus 0."},{"Start":"02:03.640 ","End":"02:09.520","Text":"In short, the answer is just 4, and we\u0027re done."}],"ID":4758},{"Watched":false,"Name":"Exercise 4","Duration":"2m 41s","ChapterTopicVideoID":4750,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this exercise, we have to find the limit as x goes to minus infinity."},{"Start":"00:04.440 ","End":"00:07.890","Text":"Again, we have a polynomial over a polynomial,"},{"Start":"00:07.890 ","End":"00:10.320","Text":"similar to the previous exercise."},{"Start":"00:10.320 ","End":"00:14.160","Text":"If we were to substitute x equals minus infinity here,"},{"Start":"00:14.160 ","End":"00:19.800","Text":"what we would get would be infinity over minus infinity,"},{"Start":"00:19.800 ","End":"00:22.260","Text":"which is another one of those forms that you can\u0027t"},{"Start":"00:22.260 ","End":"00:24.870","Text":"say what it is so this is not the approach."},{"Start":"00:24.870 ","End":"00:29.550","Text":"Instead, we\u0027re going to use the same trick as we did before by taking out"},{"Start":"00:29.550 ","End":"00:34.480","Text":"the highest power of x in the numerator and in the denominator. Let\u0027s do that."},{"Start":"00:34.480 ","End":"00:40.595","Text":"We have the limit as x tends to minus infinity."},{"Start":"00:40.595 ","End":"00:42.695","Text":"Now here in the numerator,"},{"Start":"00:42.695 ","End":"00:47.595","Text":"I can take out x^4 so I have x^4."},{"Start":"00:47.595 ","End":"00:52.040","Text":"What I\u0027m left with is 1 plus 2 over x"},{"Start":"00:52.040 ","End":"00:57.835","Text":"squared plus 6 over x^4 as the numerator."},{"Start":"00:57.835 ","End":"01:04.400","Text":"On the denominator, I can take out x cubed so I get x cubed times, here,"},{"Start":"01:04.400 ","End":"01:09.205","Text":"I have 3 plus 10 over x squared,"},{"Start":"01:09.205 ","End":"01:12.840","Text":"because x over x cubed is x squared."},{"Start":"01:12.840 ","End":"01:15.450","Text":"We can do a little bit of canceling,"},{"Start":"01:15.450 ","End":"01:21.625","Text":"the x cubed with the x^4 just leaves us with x so we just have x left here."},{"Start":"01:21.625 ","End":"01:26.145","Text":"Now, we can substitute minus infinity."},{"Start":"01:26.145 ","End":"01:28.400","Text":"From this x here,"},{"Start":"01:28.400 ","End":"01:34.880","Text":"we will have minus infinity times 1 plus 2 over minus infinity squared,"},{"Start":"01:34.880 ","End":"01:37.355","Text":"but minus infinity squared is plus infinity,"},{"Start":"01:37.355 ","End":"01:40.060","Text":"and also minus infinity to the power of 4,"},{"Start":"01:40.060 ","End":"01:45.360","Text":"an even power is also infinity, 6 over infinity."},{"Start":"01:45.360 ","End":"01:47.055","Text":"On the denominator,"},{"Start":"01:47.055 ","End":"01:50.235","Text":"left with 3 plus 10 over,"},{"Start":"01:50.235 ","End":"01:55.255","Text":"again, minus infinity squared is plus infinity so it\u0027s the 10 over infinity."},{"Start":"01:55.255 ","End":"01:58.280","Text":"I want to remind you that whenever we have"},{"Start":"01:58.280 ","End":"02:02.540","Text":"a regular finite number and we divide it by infinity,"},{"Start":"02:02.540 ","End":"02:04.295","Text":"could be plus or minus,"},{"Start":"02:04.295 ","End":"02:05.975","Text":"this is equal to 0."},{"Start":"02:05.975 ","End":"02:07.550","Text":"If I can apply that here,"},{"Start":"02:07.550 ","End":"02:09.070","Text":"here, and here,"},{"Start":"02:09.070 ","End":"02:14.870","Text":"what we\u0027d be left with is minus infinity times 1"},{"Start":"02:14.870 ","End":"02:21.635","Text":"plus 0 plus 0 over 3 plus 0,"},{"Start":"02:21.635 ","End":"02:27.720","Text":"which equals minus infinity times 1/3."},{"Start":"02:27.720 ","End":"02:30.030","Text":"This is equal to,"},{"Start":"02:30.030 ","End":"02:34.505","Text":"when we multiply minus infinity by positive quantity,"},{"Start":"02:34.505 ","End":"02:36.770","Text":"we\u0027re still left with minus infinity,"},{"Start":"02:36.770 ","End":"02:41.550","Text":"so it\u0027s just minus infinity. That\u0027s the answer."}],"ID":4759},{"Watched":false,"Name":"Exercise 5","Duration":"2m 33s","ChapterTopicVideoID":4751,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.270 ","End":"00:06.990","Text":"infinity of this expression polynomial over polynomial,"},{"Start":"00:06.990 ","End":"00:08.370","Text":"seen this thing before."},{"Start":"00:08.370 ","End":"00:13.035","Text":"The usual thing to do is to try and substitute x equals infinity,"},{"Start":"00:13.035 ","End":"00:20.070","Text":"and very quickly see that this comes out to be of the form infinity over infinity,"},{"Start":"00:20.070 ","End":"00:22.995","Text":"which is 1 of those undefined indeterminate forms."},{"Start":"00:22.995 ","End":"00:26.850","Text":"Which means that we have to use our favorite trick of"},{"Start":"00:26.850 ","End":"00:31.080","Text":"taking out the factor out of the numerator and the denominator."},{"Start":"00:31.080 ","End":"00:37.815","Text":"In this case, we get the limit as x goes to infinity."},{"Start":"00:37.815 ","End":"00:44.020","Text":"Here the highest power is x_4 so we take x_4 out of the brackets,"},{"Start":"00:44.020 ","End":"00:53.140","Text":"and we\u0027re left with 1 plus 2 over x squared plus 6 over x_4."},{"Start":"00:53.140 ","End":"00:55.485","Text":"On the denominator,"},{"Start":"00:55.485 ","End":"00:57.635","Text":"x_5 is the highest power,"},{"Start":"00:57.635 ","End":"01:01.570","Text":"will take this outside the brackets, x_5,"},{"Start":"01:01.570 ","End":"01:08.520","Text":"and we\u0027re left with 3 plus 10 over x_4."},{"Start":"01:08.520 ","End":"01:14.700","Text":"Now, we can cancel x_4 cancels because x_5 is higher so what I\u0027ll just do,"},{"Start":"01:14.700 ","End":"01:18.140","Text":"the answer will be 1 over x. I\u0027ll just cross out this,"},{"Start":"01:18.140 ","End":"01:20.460","Text":"and now cross out the 5,"},{"Start":"01:20.460 ","End":"01:23.460","Text":"which means that we have x in the denominator."},{"Start":"01:23.460 ","End":"01:28.625","Text":"What we get now we can substitute x equals infinity."},{"Start":"01:28.625 ","End":"01:34.175","Text":"What we get is 1 plus 2 over"},{"Start":"01:34.175 ","End":"01:40.325","Text":"infinity squared plus 6 over infinity to the fourth,"},{"Start":"01:40.325 ","End":"01:49.520","Text":"all over x, which is the infinity times 3 plus 10 over infinity to the fourth."},{"Start":"01:49.520 ","End":"01:53.045","Text":"Now, infinity to any power is infinity,"},{"Start":"01:53.045 ","End":"01:57.995","Text":"so all I have to remind you is that if I have some number,"},{"Start":"01:57.995 ","End":"02:00.845","Text":"regular number a over infinity,"},{"Start":"02:00.845 ","End":"02:02.915","Text":"that could be plus or minus,"},{"Start":"02:02.915 ","End":"02:04.985","Text":"that\u0027s equal to 0."},{"Start":"02:04.985 ","End":"02:07.100","Text":"In this case, this is going to be 0,"},{"Start":"02:07.100 ","End":"02:08.525","Text":"this is going to be 0."},{"Start":"02:08.525 ","End":"02:19.430","Text":"We\u0027re left with 1 plus 0 plus 0 over infinity times 3 plus 0."},{"Start":"02:19.430 ","End":"02:25.055","Text":"1 plus 0 plus 0 equals 1 and 3 times infinity,"},{"Start":"02:25.055 ","End":"02:27.935","Text":"any positive number times infinity is still infinity,"},{"Start":"02:27.935 ","End":"02:31.490","Text":"1 over infinity, and this equals 0."},{"Start":"02:31.490 ","End":"02:34.170","Text":"That\u0027s our answer. We\u0027re done."}],"ID":4760},{"Watched":false,"Name":"Exercise 6","Duration":"4m 24s","ChapterTopicVideoID":4752,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.320","Text":"In this exercise,"},{"Start":"00:01.320 ","End":"00:06.135","Text":"we have to compute the limit as x goes to infinity of this whole expression."},{"Start":"00:06.135 ","End":"00:12.135","Text":"Now, if we try to do it naively by just substituting x equals infinity,"},{"Start":"00:12.135 ","End":"00:19.710","Text":"we\u0027ll get something of the form infinity over infinity minus infinity,"},{"Start":"00:19.710 ","End":"00:22.470","Text":"which is 1 of those undefined indeterminate."},{"Start":"00:22.470 ","End":"00:24.600","Text":"Just this part alone we can\u0027t"},{"Start":"00:24.600 ","End":"00:29.610","Text":"compute and also if it was infinity minus infinity, we also couldn\u0027t do it."},{"Start":"00:29.610 ","End":"00:33.330","Text":"So we\u0027re going to have to do some algebraic manipulation for us and"},{"Start":"00:33.330 ","End":"00:37.200","Text":"our old tricks of taking out factors. Let\u0027s see."},{"Start":"00:37.200 ","End":"00:41.630","Text":"The second thing to do would be to take a factor out,"},{"Start":"00:41.630 ","End":"00:46.190","Text":"but I can\u0027t even do that because there\u0027s a difference subtraction here."},{"Start":"00:46.190 ","End":"00:50.390","Text":"The best thing is to put this whole thing over a common denominator."},{"Start":"00:50.390 ","End":"00:51.755","Text":"Let\u0027s start writing."},{"Start":"00:51.755 ","End":"00:56.705","Text":"The limit as x goes to infinity of,"},{"Start":"00:56.705 ","End":"00:58.790","Text":"let\u0027s see what a common denominator could be."},{"Start":"00:58.790 ","End":"01:01.120","Text":"Well, 2 goes into 2x plus 10."},{"Start":"01:01.120 ","End":"01:07.665","Text":"So we can just make 2x plus 10, the common denominator."},{"Start":"01:07.665 ","End":"01:10.770","Text":"Here we\u0027re left with just as it is,"},{"Start":"01:10.770 ","End":"01:18.420","Text":"x squared minus 5x plus 6 minus."},{"Start":"01:18.420 ","End":"01:21.480","Text":"But here, we\u0027re going to, because this is 2,"},{"Start":"01:21.480 ","End":"01:25.460","Text":"2 goes into 2x plus 10 this thing goes into this."},{"Start":"01:25.460 ","End":"01:26.270","Text":"I\u0027ll just write it here."},{"Start":"01:26.270 ","End":"01:29.670","Text":"X plus 5 times."},{"Start":"01:29.810 ","End":"01:34.790","Text":"Usually indicate how many times this went into the common denominator."},{"Start":"01:34.790 ","End":"01:36.755","Text":"This whole thing times 1 is what it is,"},{"Start":"01:36.755 ","End":"01:39.095","Text":"and this thing times x, x plus 5."},{"Start":"01:39.095 ","End":"01:48.930","Text":"So minus x times x plus 5 and close the brackets."},{"Start":"01:48.930 ","End":"01:53.200","Text":"Let\u0027s see, let\u0027s do the algebra and we\u0027ll get,"},{"Start":"01:53.200 ","End":"01:56.675","Text":"just scroll down a bit to give us more space."},{"Start":"01:56.675 ","End":"01:58.385","Text":"This will equal,"},{"Start":"01:58.385 ","End":"02:04.390","Text":"limit x goes to infinity of,"},{"Start":"02:04.390 ","End":"02:07.580","Text":"now let\u0027s see if we can do some of this in our heads."},{"Start":"02:07.580 ","End":"02:11.240","Text":"This thing here, this part here I\u0027ll just write it above,"},{"Start":"02:11.240 ","End":"02:18.330","Text":"is minus x squared minus 5x."},{"Start":"02:18.740 ","End":"02:23.310","Text":"If I combine the minus x squared and the x squared"},{"Start":"02:23.310 ","End":"02:27.570","Text":"cancel and the minus 5x and minus 5x is reinforced."},{"Start":"02:27.570 ","End":"02:30.100","Text":"I get minus 10x,"},{"Start":"02:30.470 ","End":"02:34.540","Text":"and then the plus 6 from here."},{"Start":"02:35.030 ","End":"02:41.920","Text":"All over 2x plus 10."},{"Start":"02:42.170 ","End":"02:44.660","Text":"I could cancel by 2,"},{"Start":"02:44.660 ","End":"02:47.585","Text":"but that\u0027s not going to help us here."},{"Start":"02:47.585 ","End":"02:49.475","Text":"The thing to do now,"},{"Start":"02:49.475 ","End":"02:53.524","Text":"is to use our trick of taking outside the brackets,"},{"Start":"02:53.524 ","End":"02:55.760","Text":"the highest power of x,"},{"Start":"02:55.760 ","End":"02:58.510","Text":"both in the numerator and in the denominator."},{"Start":"02:58.510 ","End":"03:03.815","Text":"What we get, the highest power of x is just x but if we take it out,"},{"Start":"03:03.815 ","End":"03:10.710","Text":"we get x minus 10 plus 6 over x,"},{"Start":"03:10.710 ","End":"03:13.385","Text":"and on the denominator,"},{"Start":"03:13.385 ","End":"03:20.910","Text":"we\u0027ll get x times 2 plus 10 over x."},{"Start":"03:20.910 ","End":"03:24.540","Text":"The limit as x goes to infinity."},{"Start":"03:24.540 ","End":"03:28.920","Text":"This equals, lucky that the x cancels with"},{"Start":"03:28.920 ","End":"03:33.260","Text":"the x and what we\u0027re left with if we just substitute x equals"},{"Start":"03:33.260 ","End":"03:42.670","Text":"infinity is minus 10 plus 6 over infinity"},{"Start":"03:42.670 ","End":"03:51.765","Text":"over 2 plus 10 over infinity."},{"Start":"03:51.765 ","End":"03:54.720","Text":"Now I\u0027ve mentioned that several times."},{"Start":"03:54.720 ","End":"03:56.810","Text":"I can mention it again,"},{"Start":"03:56.810 ","End":"04:02.570","Text":"that whenever we have a number and it\u0027s divided by infinity,"},{"Start":"04:02.570 ","End":"04:04.755","Text":"but it could be minus infinity,"},{"Start":"04:04.755 ","End":"04:07.110","Text":"this thing is equal to 0."},{"Start":"04:07.110 ","End":"04:15.230","Text":"I can use that both here and here and what we\u0027ll get will be minus"},{"Start":"04:15.230 ","End":"04:24.870","Text":"10 plus 0 over 2 plus 0 and that comes out to be"}],"ID":4761},{"Watched":false,"Name":"Exercise 7","Duration":"2m 30s","ChapterTopicVideoID":4753,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression."},{"Start":"00:05.280 ","End":"00:07.380","Text":"It\u0027s little bit different from before."},{"Start":"00:07.380 ","End":"00:09.765","Text":"We\u0027re used to polynomial over polynomial,"},{"Start":"00:09.765 ","End":"00:12.480","Text":"as usually you try substituting x equals infinity"},{"Start":"00:12.480 ","End":"00:15.720","Text":"first and infinity squared plus 1 is infinity."},{"Start":"00:15.720 ","End":"00:19.845","Text":"Long story short, we get something of the form infinity over infinity,"},{"Start":"00:19.845 ","End":"00:21.300","Text":"which is not good to us."},{"Start":"00:21.300 ","End":"00:26.475","Text":"We\u0027re going to have to use some small tricks algebraic manipulation."},{"Start":"00:26.475 ","End":"00:30.150","Text":"What we\u0027re going to do is something similar to what we did with polynomials."},{"Start":"00:30.150 ","End":"00:32.020","Text":"We\u0027ll take a common factor out."},{"Start":"00:32.020 ","End":"00:34.595","Text":"We\u0027ll start with under the square root sign."},{"Start":"00:34.595 ","End":"00:36.720","Text":"Let\u0027s first of all, write it,"},{"Start":"00:36.720 ","End":"00:41.975","Text":"lim as x goes to infinity of the square root,"},{"Start":"00:41.975 ","End":"00:44.840","Text":"and I\u0027ll fill this in in a minute, over x."},{"Start":"00:44.840 ","End":"00:48.580","Text":"What I\u0027m going to do is take x squared outside the brackets."},{"Start":"00:48.580 ","End":"00:51.180","Text":"I\u0027ll take x squared here,"},{"Start":"00:51.180 ","End":"00:56.045","Text":"and what I\u0027m left with is 1 minus 1 over x squared."},{"Start":"00:56.045 ","End":"00:58.730","Text":"Now, if you remember your algebra,"},{"Start":"00:58.730 ","End":"01:03.185","Text":"and I will just write down this basic algebraic fact at the side."},{"Start":"01:03.185 ","End":"01:06.530","Text":"The square root of a times b,"},{"Start":"01:06.530 ","End":"01:09.124","Text":"assuming that a and b are both positive,"},{"Start":"01:09.124 ","End":"01:14.660","Text":"is equal to the square root of a times the square root of b."},{"Start":"01:14.660 ","End":"01:16.834","Text":"If I do that here,"},{"Start":"01:16.834 ","End":"01:22.775","Text":"what I\u0027ll get is the limit x goes to infinity."},{"Start":"01:22.775 ","End":"01:26.555","Text":"Now the square root of x squared is just x,"},{"Start":"01:26.555 ","End":"01:28.325","Text":"and the square root of the other part,"},{"Start":"01:28.325 ","End":"01:29.975","Text":"I\u0027ll just write it as it is,"},{"Start":"01:29.975 ","End":"01:36.170","Text":"all over x. I\u0027ll just make another comment that the square root of x squared,"},{"Start":"01:36.170 ","End":"01:39.320","Text":"it\u0027s not always x, it\u0027s usually the absolute value of x,"},{"Start":"01:39.320 ","End":"01:43.530","Text":"but it\u0027s equal to x if x is positive."},{"Start":"01:43.530 ","End":"01:45.080","Text":"Now it\u0027s going to infinity,"},{"Start":"01:45.080 ","End":"01:46.775","Text":"so it\u0027s certainly positive."},{"Start":"01:46.775 ","End":"01:50.080","Text":"So that\u0027s why it\u0027s x and not absolute value of x."},{"Start":"01:50.080 ","End":"01:52.335","Text":"Continuing, and look at that,"},{"Start":"01:52.335 ","End":"01:54.300","Text":"we can cancel the x."},{"Start":"01:54.300 ","End":"01:59.450","Text":"At this point, we can substitute x equals infinity and get"},{"Start":"01:59.450 ","End":"02:05.800","Text":"the square root of 1 minus 1 over infinity squared."},{"Start":"02:05.800 ","End":"02:11.705","Text":"Again, I want to remind you of something that when we have a over infinity,"},{"Start":"02:11.705 ","End":"02:14.915","Text":"where a is some regular number not infinity,"},{"Start":"02:14.915 ","End":"02:17.814","Text":"this is equal to 0."},{"Start":"02:17.814 ","End":"02:20.880","Text":"Here, we have 1 over infinity."},{"Start":"02:20.880 ","End":"02:22.380","Text":"So that\u0027s 0."},{"Start":"02:22.380 ","End":"02:26.135","Text":"So this is equal to 1 minus 0, the square root of 1."},{"Start":"02:26.135 ","End":"02:28.475","Text":"The answer is just 1,"},{"Start":"02:28.475 ","End":"02:30.720","Text":"and that\u0027s the answer."}],"ID":4762},{"Watched":false,"Name":"Exercise 8","Duration":"2m 49s","ChapterTopicVideoID":4754,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.615","Text":"This exercise is very similar to the previous exercise,"},{"Start":"00:03.615 ","End":"00:05.910","Text":"where we had x going to infinity,"},{"Start":"00:05.910 ","End":"00:09.645","Text":"on here we have x tending to minus infinity."},{"Start":"00:09.645 ","End":"00:12.885","Text":"As before, if we do a straightforward substitution,"},{"Start":"00:12.885 ","End":"00:19.820","Text":"we get something of the form infinity over infinity or minus infinity."},{"Start":"00:19.820 ","End":"00:21.680","Text":"So we have to use techniques."},{"Start":"00:21.680 ","End":"00:25.340","Text":"We\u0027ll use similar techniques as before by taking x"},{"Start":"00:25.340 ","End":"00:29.095","Text":"squared outside the brackets under the square root sign."},{"Start":"00:29.095 ","End":"00:38.660","Text":"We get the limit as x goes to minus infinity of the square root of x squared,"},{"Start":"00:38.660 ","End":"00:40.580","Text":"I\u0027m taking out the brackets here,"},{"Start":"00:40.580 ","End":"00:46.060","Text":"1 plus 1 over x squared over x."},{"Start":"00:46.060 ","End":"00:48.185","Text":"Once again, like before,"},{"Start":"00:48.185 ","End":"00:49.505","Text":"I\u0027m going to split it up."},{"Start":"00:49.505 ","End":"00:54.105","Text":"The square root of a product is square root of each 1 separately."},{"Start":"00:54.105 ","End":"01:02.525","Text":"This is equal to the limit as x goes to minus infinity of the square root of"},{"Start":"01:02.525 ","End":"01:11.750","Text":"x squared times the square root of 1 plus 1 over x squared all this over x."},{"Start":"01:11.750 ","End":"01:17.725","Text":"Whereas previously, I had said that this was just x and we can cancel x with x."},{"Start":"01:17.725 ","End":"01:21.634","Text":"This time, there\u0027s a slight catch here because"},{"Start":"01:21.634 ","End":"01:27.080","Text":"the square root of x squared generally is equal to the absolute value of x."},{"Start":"01:27.080 ","End":"01:28.550","Text":"If x is negative,"},{"Start":"01:28.550 ","End":"01:31.010","Text":"the absolute value of x is minus x."},{"Start":"01:31.010 ","End":"01:36.500","Text":"In our case, the square root of x squared over x,"},{"Start":"01:36.500 ","End":"01:39.755","Text":"it\u0027s actually minus x over x,"},{"Start":"01:39.755 ","End":"01:42.140","Text":"so it\u0027s equal to minus 1."},{"Start":"01:42.140 ","End":"01:44.660","Text":"So here it doesn\u0027t exactly cancel."},{"Start":"01:44.660 ","End":"01:47.150","Text":"What it does is it gives us a minus sign."},{"Start":"01:47.150 ","End":"01:52.730","Text":"We have the limit as x goes to minus infinity"},{"Start":"01:52.730 ","End":"01:58.655","Text":"of minus the square root of 1 plus 1 over x squared."},{"Start":"01:58.655 ","End":"02:02.210","Text":"Then I also have to remind you that if a is a number then"},{"Start":"02:02.210 ","End":"02:08.155","Text":"a over infinity could be plus or minus is equal to 0."},{"Start":"02:08.155 ","End":"02:13.955","Text":"Now, we can substitute x equals minus infinity is just minus"},{"Start":"02:13.955 ","End":"02:21.105","Text":"the square root of 1 plus 1 over minus infinity squared,"},{"Start":"02:21.105 ","End":"02:24.335","Text":"and minus infinity squared is plus infinity."},{"Start":"02:24.335 ","End":"02:26.090","Text":"Either way plus or minus infinity,"},{"Start":"02:26.090 ","End":"02:28.449","Text":"1 over this thing is 0."},{"Start":"02:28.449 ","End":"02:33.675","Text":"We get minus the square root of 1 plus 0,"},{"Start":"02:33.675 ","End":"02:37.060","Text":"which is just minus 1,"},{"Start":"02:37.060 ","End":"02:39.095","Text":"and that\u0027s the answer."},{"Start":"02:39.095 ","End":"02:43.940","Text":"Just watch out for that square root of x squared when x is negative."},{"Start":"02:43.940 ","End":"02:46.460","Text":"Of course, I should have mentioned that it\u0027s negative because"},{"Start":"02:46.460 ","End":"02:50.040","Text":"it\u0027s going to minus infinity. We\u0027re done."}],"ID":4763},{"Watched":false,"Name":"Exercise 9","Duration":"2m 55s","ChapterTopicVideoID":4755,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this exercise, we have to find the limit as"},{"Start":"00:02.430 ","End":"00:05.025","Text":"x goes to minus infinity of this expression,"},{"Start":"00:05.025 ","End":"00:07.170","Text":"which has a square root in the numerator."},{"Start":"00:07.170 ","End":"00:11.610","Text":"As usual, we first try substituting minus infinity, but if you do that,"},{"Start":"00:11.610 ","End":"00:13.380","Text":"you\u0027re just going to get an expression of"},{"Start":"00:13.380 ","End":"00:17.655","Text":"the form infinity over infinity or plus or minus,"},{"Start":"00:17.655 ","End":"00:21.065","Text":"probably minus, but it\u0027s indeterminate,"},{"Start":"00:21.065 ","End":"00:24.710","Text":"undefined, and so we\u0027re going to have to use some matrix."},{"Start":"00:24.710 ","End":"00:26.090","Text":"Matrix we have used before,"},{"Start":"00:26.090 ","End":"00:29.560","Text":"which is taking out a power of x under the square root."},{"Start":"00:29.560 ","End":"00:36.215","Text":"What we\u0027ll get is the limit as x goes to minus infinity, square root,"},{"Start":"00:36.215 ","End":"00:41.385","Text":"and we\u0027ll take x^6 outside the brackets and that will leave us"},{"Start":"00:41.385 ","End":"00:47.610","Text":"with 9 minus 5x over x^6 is x^5."},{"Start":"00:47.610 ","End":"00:50.930","Text":"That\u0027s the numerator and on the denominator,"},{"Start":"00:50.930 ","End":"00:52.705","Text":"we take the x cubed out."},{"Start":"00:52.705 ","End":"00:54.799","Text":"We get x cubed,"},{"Start":"00:54.799 ","End":"01:01.100","Text":"1 minus 2 over x plus 1 over x cubed."},{"Start":"01:01.100 ","End":"01:02.960","Text":"Here something to watch out for,"},{"Start":"01:02.960 ","End":"01:04.070","Text":"we are going to of course,"},{"Start":"01:04.070 ","End":"01:11.495","Text":"use the formula where the square root of ab equals square root of a,"},{"Start":"01:11.495 ","End":"01:14.210","Text":"square root of b and in this case,"},{"Start":"01:14.210 ","End":"01:17.120","Text":"we\u0027re going to have to deal with the square root of x^6,"},{"Start":"01:17.120 ","End":"01:21.170","Text":"and normally the square root of something squared,"},{"Start":"01:21.170 ","End":"01:26.660","Text":"this is the square root of x cubed squared."},{"Start":"01:26.660 ","End":"01:28.880","Text":"You would think it would be x cubed,"},{"Start":"01:28.880 ","End":"01:32.420","Text":"but you should remember that it\u0027s absolute value of x cubed."},{"Start":"01:32.420 ","End":"01:34.880","Text":"Now, x is going to minus infinity,"},{"Start":"01:34.880 ","End":"01:36.785","Text":"so it certainly negative."},{"Start":"01:36.785 ","End":"01:40.850","Text":"So x cubed is also negative because negative cubed is negative."},{"Start":"01:40.850 ","End":"01:44.690","Text":"What this means that the absolute value of a negative quantity"},{"Start":"01:44.690 ","End":"01:48.890","Text":"is its negation is this is equal to minus x cubed."},{"Start":"01:48.890 ","End":"01:51.050","Text":"With this in mind,"},{"Start":"01:51.050 ","End":"01:55.340","Text":"limit as x goes to minus infinity,"},{"Start":"01:55.340 ","End":"02:01.355","Text":"we get the square root of x^6 over x cubed."},{"Start":"02:01.355 ","End":"02:10.760","Text":"That continuing, we also get the square root of 9 minus 5 over x^5 times this thing."},{"Start":"02:10.760 ","End":"02:15.770","Text":"Now, this over this doesn\u0027t exactly cancel because what we saw there,"},{"Start":"02:15.770 ","End":"02:17.975","Text":"but we do get a minus."},{"Start":"02:17.975 ","End":"02:21.770","Text":"The other thing I wanted to remind you of is that if we have a number a"},{"Start":"02:21.770 ","End":"02:25.670","Text":"over whether it\u0027s plus infinity or minus infinity,"},{"Start":"02:25.670 ","End":"02:27.230","Text":"this is equal to 0."},{"Start":"02:27.230 ","End":"02:29.630","Text":"We\u0027re going to use that here, here and here."},{"Start":"02:29.630 ","End":"02:34.790","Text":"What we end up getting is this is equal to minus, from here,"},{"Start":"02:34.790 ","End":"02:43.580","Text":"the square root of 9 minus 0 over 1 minus 0 plus 0."},{"Start":"02:43.580 ","End":"02:46.160","Text":"In short, the square root of 9 is 3,"},{"Start":"02:46.160 ","End":"02:52.760","Text":"the denominator\u0027s 1 minus 3 over 1 is just minus 3."},{"Start":"02:52.760 ","End":"02:55.440","Text":"That\u0027s all there is to it."}],"ID":4764},{"Watched":false,"Name":"Exercise 10","Duration":"4m 47s","ChapterTopicVideoID":4756,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.735","Text":"In this exercise, we have to find the limit as x goes to infinity"},{"Start":"00:03.735 ","End":"00:08.760","Text":"of the cube root of some polynomial over the square root of another polynomial."},{"Start":"00:08.760 ","End":"00:11.535","Text":"We\u0027re going to use our usual bag of tricks,"},{"Start":"00:11.535 ","End":"00:16.410","Text":"which is taking out under the root sign the highest power of x."},{"Start":"00:16.410 ","End":"00:25.350","Text":"Let\u0027s just rewrite this as the limit as x goes to infinity,"},{"Start":"00:25.350 ","End":"00:30.300","Text":"and the numerator, the cube root of,"},{"Start":"00:30.300 ","End":"00:32.895","Text":"now the highest power of x, not the first one,"},{"Start":"00:32.895 ","End":"00:37.920","Text":"is x^6, so let\u0027s write down x^6,"},{"Start":"00:37.920 ","End":"00:41.285","Text":"and this, we have to take outside the brackets."},{"Start":"00:41.285 ","End":"00:50.555","Text":"What we\u0027re left with is x^4 over x^6 is 1 over x squared."},{"Start":"00:50.555 ","End":"00:59.120","Text":"Here taking out x^6 gives us 2 over x^4."},{"Start":"00:59.120 ","End":"01:07.480","Text":"Here 6 over x^6, and the last one,"},{"Start":"01:07.480 ","End":"01:10.440","Text":"just 27 on its own,"},{"Start":"01:10.440 ","End":"01:12.815","Text":"and now the denominator,"},{"Start":"01:12.815 ","End":"01:17.930","Text":"the square root, the highest power here is x^4,"},{"Start":"01:17.930 ","End":"01:21.470","Text":"so it\u0027s x^4,"},{"Start":"01:21.470 ","End":"01:23.985","Text":"and when we take it out,"},{"Start":"01:23.985 ","End":"01:29.805","Text":"we get 3x cubed over x^4 is 3 over x."},{"Start":"01:29.805 ","End":"01:34.575","Text":"10 is going to be over x cubed,"},{"Start":"01:34.575 ","End":"01:38.080","Text":"and here, just 4 on its own."},{"Start":"01:38.080 ","End":"01:42.355","Text":"Now I want to remind you again that we are going to use some formulas,"},{"Start":"01:42.355 ","End":"01:52.105","Text":"1 is that the square root of ab is the square root of a square root of b,"},{"Start":"01:52.105 ","End":"01:55.210","Text":"and that\u0027s provided that a and b are positive,"},{"Start":"01:55.210 ","End":"02:01.030","Text":"and if they\u0027re positive, the same thing holds for the cube root of ab."},{"Start":"02:01.030 ","End":"02:04.900","Text":"Let\u0027s just write that down here."},{"Start":"02:04.900 ","End":"02:08.980","Text":"There\u0027s a restriction that a and b are positive for the cube root of a b is always"},{"Start":"02:08.980 ","End":"02:13.360","Text":"equal to the cube root of a cube root of b,"},{"Start":"02:13.360 ","End":"02:16.090","Text":"whether a and b are positive or not."},{"Start":"02:16.090 ","End":"02:19.915","Text":"So that\u0027s 1 thing. I know that we\u0027re also going to use the fact that"},{"Start":"02:19.915 ","End":"02:26.005","Text":"a number over plus or minus infinity is 0."},{"Start":"02:26.005 ","End":"02:34.975","Text":"What we get is the limit x goes to infinity,"},{"Start":"02:34.975 ","End":"02:38.120","Text":"we get the cube root"},{"Start":"02:38.120 ","End":"02:48.720","Text":"of x^6 times the cube root of all the rest of it,"},{"Start":"02:48.720 ","End":"03:00.165","Text":"which is 1 over x squared plus 2 over x^4 plus 6 over x^6 plus 27."},{"Start":"03:00.165 ","End":"03:10.900","Text":"On the denominator, the square root of x^4 times the square root of,"},{"Start":"03:10.900 ","End":"03:14.605","Text":"here I forgot to close the bracket."},{"Start":"03:14.605 ","End":"03:19.720","Text":"Here we get the 3 over x"},{"Start":"03:19.850 ","End":"03:27.075","Text":"plus 10 over x cubed plus 4."},{"Start":"03:27.075 ","End":"03:31.670","Text":"Now at the side here what I\u0027m going to do,"},{"Start":"03:31.670 ","End":"03:36.980","Text":"the cube root of x^6 is always equal"},{"Start":"03:36.980 ","End":"03:43.070","Text":"to x^6 over 3 it\u0027s x squared."},{"Start":"03:43.070 ","End":"03:49.010","Text":"The square root of x^4 is x squared squared,"},{"Start":"03:49.010 ","End":"03:54.725","Text":"is actually equal to the absolute value of x squared,"},{"Start":"03:54.725 ","End":"03:59.300","Text":"but since x is turning to infinity it\u0027s positive,"},{"Start":"03:59.300 ","End":"04:02.720","Text":"so it\u0027s actually equal to x squared."},{"Start":"04:02.720 ","End":"04:06.680","Text":"What we can do here is just cancel these 2 things."},{"Start":"04:06.680 ","End":"04:09.905","Text":"This is x squared and this is x squared."},{"Start":"04:09.905 ","End":"04:15.379","Text":"The next thing we\u0027ll do is use this fact that a number over infinity is 0."},{"Start":"04:15.379 ","End":"04:18.110","Text":"That will give us 0 here, here,"},{"Start":"04:18.110 ","End":"04:21.805","Text":"and here, and here, and here."},{"Start":"04:21.805 ","End":"04:25.415","Text":"What we\u0027re left with if we substitute x is infinity,"},{"Start":"04:25.415 ","End":"04:30.225","Text":"is just the cube root of"},{"Start":"04:30.225 ","End":"04:37.340","Text":"27 over the square root of 4."},{"Start":"04:37.340 ","End":"04:40.565","Text":"Cube root of 27 is 3,"},{"Start":"04:40.565 ","End":"04:42.560","Text":"square root of 4 is 2,"},{"Start":"04:42.560 ","End":"04:47.670","Text":"the answer is 3 over 2, and we\u0027re done."}],"ID":4765},{"Watched":false,"Name":"Exercise 11","Duration":"4m 20s","ChapterTopicVideoID":4757,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.300 ","End":"00:06.885","Text":"infinity of this expression involving square roots."},{"Start":"00:06.885 ","End":"00:10.830","Text":"Notice that all the expressions under the square root are linear."},{"Start":"00:10.830 ","End":"00:13.650","Text":"They all have the highest power is x."},{"Start":"00:13.650 ","End":"00:16.920","Text":"The usual technique of putting x equals infinity won\u0027t work."},{"Start":"00:16.920 ","End":"00:20.190","Text":"Infinity minus infinity is undefined and so on."},{"Start":"00:20.190 ","End":"00:24.900","Text":"We\u0027re going to have to use our familiar algebraic tricks,"},{"Start":"00:24.900 ","End":"00:32.040","Text":"and we\u0027re going to write this as is the limit of course x goes to infinity."},{"Start":"00:32.040 ","End":"00:35.555","Text":"In each case, the highest power of x is just x itself."},{"Start":"00:35.555 ","End":"00:44.045","Text":"We\u0027ll write it out as the square root of x times 1 plus 2 over x"},{"Start":"00:44.045 ","End":"00:54.785","Text":"minus the square root of x times 3 minus 3 over x."},{"Start":"00:54.785 ","End":"00:58.115","Text":"That\u0027s the numerator, the denominator we have."},{"Start":"00:58.115 ","End":"01:00.320","Text":"Again, we take x outside the brackets,"},{"Start":"01:00.320 ","End":"01:05.040","Text":"we\u0027re left with 4 plus 1 over x."},{"Start":"01:05.040 ","End":"01:15.905","Text":"Here we get the square root of x times 5 minus 1 over x."},{"Start":"01:15.905 ","End":"01:21.160","Text":"What I\u0027m going to do, we\u0027re going to use our usual formula, for example,"},{"Start":"01:21.160 ","End":"01:27.175","Text":"that the square root of ab is equal to the square root of a,"},{"Start":"01:27.175 ","End":"01:28.570","Text":"square root of b,"},{"Start":"01:28.570 ","End":"01:31.000","Text":"provided a and b are positive."},{"Start":"01:31.000 ","End":"01:40.605","Text":"The other formula that we\u0027re going to use is that a number a divided by infinity is 0."},{"Start":"01:40.605 ","End":"01:49.105","Text":"What we get is the limit x goes to infinity of the square root of x,"},{"Start":"01:49.105 ","End":"01:58.440","Text":"square root of 1 plus 2 over x minus the square root of x,"},{"Start":"01:58.440 ","End":"01:59.920","Text":"oh, here forgotten, the bracket,"},{"Start":"01:59.920 ","End":"02:05.780","Text":"times the square root of 3 minus 3 over x."},{"Start":"02:05.780 ","End":"02:10.189","Text":"All this over square root of x,"},{"Start":"02:10.189 ","End":"02:15.440","Text":"square root of 4 plus 1 over x minus finally,"},{"Start":"02:15.440 ","End":"02:21.500","Text":"square root of x, square root of 5 minus 1 over x."},{"Start":"02:21.500 ","End":"02:23.285","Text":"Now if you notice,"},{"Start":"02:23.285 ","End":"02:26.420","Text":"we have the square root of x in a lot of places."},{"Start":"02:26.420 ","End":"02:30.470","Text":"Now, if I took the square root of x outside the brackets,"},{"Start":"02:30.470 ","End":"02:33.440","Text":"I get a difference here of these 2 and the difference"},{"Start":"02:33.440 ","End":"02:36.935","Text":"here of these 2 and I could actually cancel top and bottom."},{"Start":"02:36.935 ","End":"02:43.070","Text":"But those of you who are a little bit rusty with your algebra, I\u0027ll just say that,"},{"Start":"02:43.070 ","End":"02:53.290","Text":"this looks something like ab minus ac over ad minus ae."},{"Start":"02:53.290 ","End":"02:56.600","Text":"What we would do, we take a outside the brackets"},{"Start":"02:56.600 ","End":"02:58.310","Text":"and get b minus c."},{"Start":"02:58.310 ","End":"03:00.050","Text":"Here we take a outside"},{"Start":"03:00.050 ","End":"03:03.980","Text":"the brackets and get d minus e."},{"Start":"03:03.980 ","End":"03:06.965","Text":"You can easily see that we would cancel the a."},{"Start":"03:06.965 ","End":"03:10.910","Text":"We might as well have canceled it in the first place over here."},{"Start":"03:10.910 ","End":"03:13.910","Text":"This equals limit x goes to infinity."},{"Start":"03:13.910 ","End":"03:15.770","Text":"Like I said, I\u0027m canceling this,"},{"Start":"03:15.770 ","End":"03:17.450","Text":"which is not the normal cancellation,"},{"Start":"03:17.450 ","End":"03:19.805","Text":"but here\u0027s my justification."},{"Start":"03:19.805 ","End":"03:25.400","Text":"Just write it as square root of 1 plus 2"},{"Start":"03:25.400 ","End":"03:31.600","Text":"over x minus the square root of 3 minus 3 over x."},{"Start":"03:31.600 ","End":"03:34.560","Text":"The stuff keeps repeating,"},{"Start":"03:34.560 ","End":"03:39.255","Text":"the set 5 minus 1 over x."},{"Start":"03:39.255 ","End":"03:43.645","Text":"Now if I put x is as infinity,"},{"Start":"03:43.645 ","End":"03:47.690","Text":"this is the formula that a over infinity is 0."},{"Start":"03:47.690 ","End":"03:49.670","Text":"I\u0027m going to get a 0 here."},{"Start":"03:49.670 ","End":"03:54.440","Text":"I\u0027m going to get a 0 here and here, and here."},{"Start":"03:54.440 ","End":"04:02.360","Text":"What I\u0027m going to be left with is from here I\u0027ll get a square root of 1."},{"Start":"04:02.360 ","End":"04:06.350","Text":"From here I get square root of 3."},{"Start":"04:06.350 ","End":"04:09.245","Text":"I\u0027ll copy the minus."},{"Start":"04:09.245 ","End":"04:13.380","Text":"From here I get the square root of 5."},{"Start":"04:13.420 ","End":"04:17.180","Text":"From here I get the square root of 4."},{"Start":"04:17.180 ","End":"04:20.549","Text":"This basically is our answer."}],"ID":4766},{"Watched":false,"Name":"Exercise 12","Duration":"6m 32s","ChapterTopicVideoID":4758,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we have to find the limit as"},{"Start":"00:04.050 ","End":"00:08.055","Text":"x goes to minus infinity of this whole expression."},{"Start":"00:08.055 ","End":"00:12.900","Text":"This is very similar to the previous exercise only there we had infinity."},{"Start":"00:12.900 ","End":"00:16.020","Text":"The first thing to do with infinity is to try and substitute."},{"Start":"00:16.020 ","End":"00:20.025","Text":"I\u0027d like to just remind you of a formula."},{"Start":"00:20.025 ","End":"00:27.855","Text":"The first one is that a^minus infinity is equal to 0,"},{"Start":"00:27.855 ","End":"00:32.445","Text":"provided that a is bigger than 1."},{"Start":"00:32.445 ","End":"00:36.980","Text":"If we try substituting x is minus infinity,"},{"Start":"00:36.980 ","End":"00:40.640","Text":"these exponents are all minus infinity also."},{"Start":"00:40.640 ","End":"00:45.665","Text":"According to this, it will be 0 so we get 0 plus 0 plus 0 plus 0."},{"Start":"00:45.665 ","End":"00:50.810","Text":"In short, we get one of those undefined indeterminate forms,"},{"Start":"00:50.810 ","End":"00:52.290","Text":"0 over 0,"},{"Start":"00:52.290 ","End":"00:58.295","Text":"so we\u0027ll have to use some algebraic techniques to simplify this."},{"Start":"00:58.295 ","End":"00:59.970","Text":"This is a problem with exponents,"},{"Start":"00:59.970 ","End":"01:03.560","Text":"so you might as well remember some of the rules for the exponents."},{"Start":"01:03.560 ","End":"01:08.160","Text":"The main ones are that a^b plus"},{"Start":"01:08.160 ","End":"01:14.650","Text":"c is actually the product a^b times a^c."},{"Start":"01:14.650 ","End":"01:18.500","Text":"The other formula that\u0027s very similar is with a minus,"},{"Start":"01:18.500 ","End":"01:23.975","Text":"in which case we get a^b over a^c."},{"Start":"01:23.975 ","End":"01:28.530","Text":"The third useful formula is a power of a power,"},{"Start":"01:29.260 ","End":"01:35.360","Text":"a^b^c is just a^bc."},{"Start":"01:35.360 ","End":"01:37.760","Text":"We have different bases here."},{"Start":"01:37.760 ","End":"01:38.960","Text":"This is a base 2,"},{"Start":"01:38.960 ","End":"01:41.420","Text":"base 2, base 4, base 16,"},{"Start":"01:41.420 ","End":"01:49.880","Text":"but we could convert them all to base 2 because 16 is 2^4,"},{"Start":"01:49.880 ","End":"01:53.970","Text":"and 4 is 2^2."},{"Start":"01:54.320 ","End":"01:57.765","Text":"We could rewrite this limit,"},{"Start":"01:57.765 ","End":"02:00.470","Text":"x goes to"},{"Start":"02:00.470 ","End":"02:08.930","Text":"minus infinity of 2^4^x."},{"Start":"02:08.930 ","End":"02:10.010","Text":"I use this formula,"},{"Start":"02:10.010 ","End":"02:12.565","Text":"so it\u0027s 4 times x."},{"Start":"02:12.565 ","End":"02:18.630","Text":"The second one here is 2^2^x plus 1,"},{"Start":"02:18.630 ","End":"02:21.015","Text":"so it\u0027s 2 times x plus 1."},{"Start":"02:21.015 ","End":"02:22.470","Text":"Twice x plus 1,"},{"Start":"02:22.470 ","End":"02:25.545","Text":"I\u0027ll write it straight away as 2x plus 2."},{"Start":"02:25.545 ","End":"02:31.200","Text":"Denominator, 2^4x plus"},{"Start":"02:31.200 ","End":"02:38.090","Text":"2 as is and 2^x plus 3 as is."},{"Start":"02:38.090 ","End":"02:40.700","Text":"Now in case you did the previous exercise,"},{"Start":"02:40.700 ","End":"02:44.540","Text":"you\u0027ll note that there we took 2^4x outside the brackets,"},{"Start":"02:44.540 ","End":"02:46.190","Text":"and that seemed to work for us,"},{"Start":"02:46.190 ","End":"02:47.735","Text":"here it won\u0027t work."},{"Start":"02:47.735 ","End":"02:51.230","Text":"Trial and error shows that you should actually take this outside"},{"Start":"02:51.230 ","End":"02:55.355","Text":"the brackets here and this also in the denominator."},{"Start":"02:55.355 ","End":"02:58.160","Text":"In this case, what\u0027s going to work is taking this outside"},{"Start":"02:58.160 ","End":"03:02.090","Text":"the brackets in the numerator and here in the denominator."},{"Start":"03:02.090 ","End":"03:08.925","Text":"Continuing, we get the limit as x goes to minus infinity."},{"Start":"03:08.925 ","End":"03:15.000","Text":"Like I said, here we take 2x plus 2 outside the brackets"},{"Start":"03:15.000 ","End":"03:21.900","Text":"and here I\u0027ll take 2^x plus 3 outside the brackets."},{"Start":"03:21.900 ","End":"03:24.660","Text":"We have to do here,"},{"Start":"03:24.660 ","End":"03:29.055","Text":"2_4x over 2^2x plus 2."},{"Start":"03:29.055 ","End":"03:36.800","Text":"We need to use this formula and do here we have 2^4x"},{"Start":"03:36.800 ","End":"03:41.660","Text":"over 2^2x plus 2"},{"Start":"03:41.660 ","End":"03:48.739","Text":"is 2^4x minus 2x plus 2,"},{"Start":"03:48.739 ","End":"03:57.840","Text":"which is 2^4x minus 2x is 2x and minus the 2."},{"Start":"03:57.840 ","End":"04:04.980","Text":"So here we have 2^2x minus 2."},{"Start":"04:04.980 ","End":"04:07.570","Text":"The next one is just 1."},{"Start":"04:07.570 ","End":"04:09.110","Text":"Here we have a similar thing."},{"Start":"04:09.110 ","End":"04:13.970","Text":"We have to do, 2^4x plus"},{"Start":"04:13.970 ","End":"04:19.490","Text":"2 over 2^x plus 3,"},{"Start":"04:19.490 ","End":"04:28.850","Text":"which is 2^4x plus 2 less x plus 3,"},{"Start":"04:28.850 ","End":"04:34.530","Text":"which is 2^4x minus x is 3x,"},{"Start":"04:34.530 ","End":"04:39.730","Text":"2 minus 3 is minus 1."},{"Start":"04:39.730 ","End":"04:47.190","Text":"Here we get 2^3x minus 1."},{"Start":"04:47.190 ","End":"04:49.950","Text":"Here we just get 1."},{"Start":"04:49.950 ","End":"04:52.100","Text":"Now we do a little bit of canceling,"},{"Start":"04:52.100 ","End":"04:53.210","Text":"this over this,"},{"Start":"04:53.210 ","End":"04:57.300","Text":"we use the formula that this over this equals this."},{"Start":"04:57.300 ","End":"05:01.740","Text":"We have to do 2x plus 2 less x plus 3,"},{"Start":"05:01.740 ","End":"05:11.295","Text":"and that will give us 2x minus x is x and 2 minus 3 is minus 1."},{"Start":"05:11.295 ","End":"05:14.910","Text":"I\u0027ll put that outside the fraction."},{"Start":"05:14.910 ","End":"05:22.485","Text":"Here we get 2^2x minus 2 plus"},{"Start":"05:22.485 ","End":"05:32.135","Text":"1 over 2^3x minus 1 plus 1."},{"Start":"05:32.135 ","End":"05:36.545","Text":"Now notice that when x goes to minus infinity,"},{"Start":"05:36.545 ","End":"05:41.405","Text":"that all these expressions also go to minus infinity."},{"Start":"05:41.405 ","End":"05:46.585","Text":"Minus infinity takeaway 1 is minus infinity twice minus infinity,"},{"Start":"05:46.585 ","End":"05:49.290","Text":"because 2 is positive is also minus infinity,"},{"Start":"05:49.290 ","End":"05:53.615","Text":"and the numbers minus 2 doesn\u0027t change it and this is minus infinity."},{"Start":"05:53.615 ","End":"05:58.670","Text":"Basically all these things are minus infinity and we\u0027re going to use"},{"Start":"05:58.670 ","End":"06:04.805","Text":"this formula because 2 is bigger than 1 so the power of minus infinity is 0."},{"Start":"06:04.805 ","End":"06:07.220","Text":"It\u0027s equal when we put minus infinity in,"},{"Start":"06:07.220 ","End":"06:10.475","Text":"2^minus infinity is 0,"},{"Start":"06:10.475 ","End":"06:12.240","Text":"this would be 0,"},{"Start":"06:12.240 ","End":"06:15.105","Text":"this is plus 1,"},{"Start":"06:15.105 ","End":"06:18.545","Text":"this is also 0 plus 1."},{"Start":"06:18.545 ","End":"06:20.760","Text":"Anyway, 0 times something,"},{"Start":"06:20.760 ","End":"06:24.725","Text":"as long as this is not 0 in the denominator,"},{"Start":"06:24.725 ","End":"06:32.160","Text":"the whole thing comes out to be just equal to 0 and that\u0027s the answer."}],"ID":4767},{"Watched":false,"Name":"Exercise 13","Duration":"4m 59s","ChapterTopicVideoID":4759,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.565","Text":"In this exercise, we have to find the limit of x goes to infinity of this expression,"},{"Start":"00:05.565 ","End":"00:09.120","Text":"which involves the exponents and the fraction."},{"Start":"00:09.120 ","End":"00:12.015","Text":"We\u0027ve seen this exercise before,"},{"Start":"00:12.015 ","End":"00:20.115","Text":"and the direct substitution of infinity just leads us to infinity over infinity,"},{"Start":"00:20.115 ","End":"00:23.070","Text":"so we can\u0027t just substitute,"},{"Start":"00:23.070 ","End":"00:26.895","Text":"we\u0027re going to have to do some algebra here and simplify."},{"Start":"00:26.895 ","End":"00:30.375","Text":"I\u0027m going to remind you of some of the basic rules of"},{"Start":"00:30.375 ","End":"00:34.125","Text":"exponents that you should be pretty familiar with them by now."},{"Start":"00:34.125 ","End":"00:44.230","Text":"That a^b plus c is a^b times a^c."},{"Start":"00:44.230 ","End":"00:47.255","Text":"Also, if we have a minus here,"},{"Start":"00:47.255 ","End":"00:53.710","Text":"it turns into a quotient, a^b over a^c."},{"Start":"00:53.710 ","End":"00:58.610","Text":"If we have an exponent of an exponent a^b^c,"},{"Start":"00:58.610 ","End":"01:04.140","Text":"that\u0027s a to the power of bc."},{"Start":"01:04.140 ","End":"01:06.920","Text":"If we notice the basis here are 3,"},{"Start":"01:06.920 ","End":"01:09.335","Text":"3, 9 and 81,"},{"Start":"01:09.335 ","End":"01:14.360","Text":"all of these can be expressed in terms of base 3 because the"},{"Start":"01:14.360 ","End":"01:22.875","Text":"9 is just 3 squared and the 81 is 3 to the fourth."},{"Start":"01:22.875 ","End":"01:27.230","Text":"So if I use these and substitute in here,"},{"Start":"01:27.230 ","End":"01:33.095","Text":"what I will get is the limit x goes to infinity."},{"Start":"01:33.095 ","End":"01:35.220","Text":"The 4 is just there."},{"Start":"01:35.220 ","End":"01:39.620","Text":"9^x is 3^2^x."},{"Start":"01:39.620 ","End":"01:44.670","Text":"Using this formula, next 3, I\u0027ll leave alone."},{"Start":"01:44.670 ","End":"01:46.995","Text":"X plus 1,"},{"Start":"01:46.995 ","End":"01:49.980","Text":"81 is 3 to the fourth,"},{"Start":"01:49.980 ","End":"01:53.235","Text":"and it\u0027s to the power of 0.5x,"},{"Start":"01:53.235 ","End":"01:58.770","Text":"so I get 3^4"},{"Start":"01:58.770 ","End":"02:05.190","Text":"times 0.5x."},{"Start":"02:05.190 ","End":"02:07.380","Text":"Here 3 is what we want,"},{"Start":"02:07.380 ","End":"02:12.574","Text":"so it just leaves it, 3^x plus 3."},{"Start":"02:12.574 ","End":"02:16.055","Text":"The next step is to take something outside the brackets,"},{"Start":"02:16.055 ","End":"02:19.580","Text":"and we behave differently with infinity as with minus infinity."},{"Start":"02:19.580 ","End":"02:21.125","Text":"Where it\u0027s plus infinity,"},{"Start":"02:21.125 ","End":"02:28.515","Text":"we generally take the larger of the coefficients on the 2x and the x plus 1,"},{"Start":"02:28.515 ","End":"02:30.030","Text":"the 2 is larger,"},{"Start":"02:30.030 ","End":"02:33.430","Text":"so that\u0027s what we take out is 3^2x."},{"Start":"02:33.430 ","End":"02:38.580","Text":"Here I\u0027ll just point out to the side here that 4 times"},{"Start":"02:38.580 ","End":"02:44.090","Text":"0.5 is 2 instead of that, write 2x."},{"Start":"02:44.090 ","End":"02:46.790","Text":"Here we have 2x and x plus 3, again,"},{"Start":"02:46.790 ","End":"02:51.035","Text":"we\u0027ll choose the 2x to take outside the brackets."},{"Start":"02:51.035 ","End":"02:59.480","Text":"What we get is the limit as x goes to infinity."},{"Start":"02:59.480 ","End":"03:03.860","Text":"Now here we take 3^2x outside,"},{"Start":"03:03.860 ","End":"03:09.770","Text":"and in the denominator also 3^2x we take outside the brackets."},{"Start":"03:09.770 ","End":"03:11.135","Text":"Let\u0027s see what we\u0027re left with."},{"Start":"03:11.135 ","End":"03:12.760","Text":"Here we have a 4,"},{"Start":"03:12.760 ","End":"03:16.580","Text":"here we use this rule here of the subtraction,"},{"Start":"03:16.580 ","End":"03:22.610","Text":"we need x plus 1 minus 2x is 1 minus x."},{"Start":"03:22.610 ","End":"03:30.695","Text":"So that\u0027s 3^1 minus x."},{"Start":"03:30.695 ","End":"03:34.820","Text":"Now the denominator, we\u0027ve taken out 3^2x,"},{"Start":"03:34.820 ","End":"03:37.315","Text":"so here we\u0027re just left with 1,"},{"Start":"03:37.315 ","End":"03:40.905","Text":"and if we take out 2x from here,"},{"Start":"03:40.905 ","End":"03:51.130","Text":"so it\u0027s 3 instead of 1 minus x here we have 3 minus x, and this cancels."},{"Start":"03:52.040 ","End":"03:58.435","Text":"This point, all we have to do is substitute x equals infinity,"},{"Start":"03:58.435 ","End":"04:00.670","Text":"and what we get is,"},{"Start":"04:00.670 ","End":"04:09.190","Text":"from here, this 4 is this 4, 3^1 minus infinity."},{"Start":"04:09.190 ","End":"04:13.180","Text":"From the denominator, we get a 1, just copy it,"},{"Start":"04:13.180 ","End":"04:19.150","Text":"and 3^3 minus infinity."},{"Start":"04:19.150 ","End":"04:23.750","Text":"Another thing that I should have mentioned is that a to"},{"Start":"04:23.750 ","End":"04:29.330","Text":"the power of minus infinity is equal to 0,"},{"Start":"04:29.330 ","End":"04:31.610","Text":"provided that a is bigger than 1."},{"Start":"04:31.610 ","End":"04:33.815","Text":"For example, if a is 3,"},{"Start":"04:33.815 ","End":"04:40.310","Text":"1 minus infinity is minus infinity because adding the 1 doesn\u0027t change an infinity."},{"Start":"04:40.310 ","End":"04:44.045","Text":"Same here, 3 to the minus infinity."},{"Start":"04:44.045 ","End":"04:53.730","Text":"What we get is just 4 plus 0 over 1 plus 0,"},{"Start":"04:53.730 ","End":"04:57.750","Text":"and that boils down to just 4."},{"Start":"04:57.750 ","End":"05:00.160","Text":"That\u0027s our answer."}],"ID":4768},{"Watched":false,"Name":"Exercise 14","Duration":"4m 44s","ChapterTopicVideoID":4760,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.565","Text":"This exercise is the same as the previous one,"},{"Start":"00:03.565 ","End":"00:08.340","Text":"except that we have a minus infinity instead of infinity."},{"Start":"00:08.340 ","End":"00:11.650","Text":"So we will reuse some of the techniques of the previous exercise,"},{"Start":"00:11.650 ","End":"00:14.015","Text":"but there will also be differences."},{"Start":"00:14.015 ","End":"00:16.250","Text":"The first thing I\u0027d like to mention,"},{"Start":"00:16.250 ","End":"00:19.570","Text":"because we\u0027re going to try and substitute x equals minus infinity,"},{"Start":"00:19.570 ","End":"00:25.665","Text":"is that a to the power of minus infinity is 0, so I emphasize."},{"Start":"00:25.665 ","End":"00:27.400","Text":"The other thing is that we\u0027re going to use"},{"Start":"00:27.400 ","End":"00:30.595","Text":"the same trick as before of using the common base 3."},{"Start":"00:30.595 ","End":"00:38.860","Text":"We can do this because 9 is 3 to the power of 2 and 81 is 3 to the power of 4."},{"Start":"00:38.860 ","End":"00:43.000","Text":"Let\u0027s rewrite this limit in terms of base 3."},{"Start":"00:43.000 ","End":"00:49.335","Text":"Limit x goes to minus infinity."},{"Start":"00:49.335 ","End":"00:54.270","Text":"4 time 9 is 3 squared so the 2 goes"},{"Start":"00:54.270 ","End":"00:59.970","Text":"with the x so we get 3 to the power of 2x."},{"Start":"00:59.970 ","End":"01:09.450","Text":"Here, just the same 3 to the x plus 1 divided by 81 is 3 to the 4,"},{"Start":"01:09.450 ","End":"01:12.910","Text":"so we need to multiply 4 times this thing."},{"Start":"01:12.910 ","End":"01:19.845","Text":"Well, 4 times a 1/2 is 2 so we get 3 to the 2x here,"},{"Start":"01:19.845 ","End":"01:21.885","Text":"and this stays the same,"},{"Start":"01:21.885 ","End":"01:25.615","Text":"3 to the x plus 3."},{"Start":"01:25.615 ","End":"01:27.770","Text":"Now, the difference between this and"},{"Start":"01:27.770 ","End":"01:30.755","Text":"the previous exercise is what we take outside the bracket."},{"Start":"01:30.755 ","End":"01:32.465","Text":"In the case of infinity,"},{"Start":"01:32.465 ","End":"01:35.225","Text":"it\u0027s the 2x that\u0027s chosen,"},{"Start":"01:35.225 ","End":"01:37.670","Text":"basically the one with the larger coefficient."},{"Start":"01:37.670 ","End":"01:39.815","Text":"But in the case of minus infinity,"},{"Start":"01:39.815 ","End":"01:42.035","Text":"we\u0027re going to choose the smaller one."},{"Start":"01:42.035 ","End":"01:46.640","Text":"We\u0027re going to choose here the x plus 1 and here the x plus 3."},{"Start":"01:46.640 ","End":"01:55.040","Text":"That may make this more precise limit x goes to minus infinity."},{"Start":"01:55.040 ","End":"02:02.050","Text":"Now at the top, we take out 3 to the x plus 1,"},{"Start":"02:02.050 ","End":"02:03.584","Text":"and at the bottom,"},{"Start":"02:03.584 ","End":"02:08.910","Text":"3 to the x plus 3,"},{"Start":"02:08.910 ","End":"02:11.540","Text":"and then inside the brackets,"},{"Start":"02:11.540 ","End":"02:13.370","Text":"the easy part is the last one,"},{"Start":"02:13.370 ","End":"02:16.170","Text":"here it\u0027s going to be plus 1 and"},{"Start":"02:16.170 ","End":"02:19.745","Text":"here it\u0027s going to be plus 1 because of what we took out."},{"Start":"02:19.745 ","End":"02:25.100","Text":"What we\u0027re left with is 3 to the 2x."},{"Start":"02:25.100 ","End":"02:28.730","Text":"We need to divide it by what we took out,"},{"Start":"02:28.730 ","End":"02:32.435","Text":"which is 3 to the x plus 1,"},{"Start":"02:32.435 ","End":"02:35.420","Text":"and that\u0027s 3 to the power of,"},{"Start":"02:35.420 ","End":"02:40.985","Text":"subtract 2x minus x plus 1,"},{"Start":"02:40.985 ","End":"02:47.120","Text":"which is 3 to the x minus 1."},{"Start":"02:47.120 ","End":"02:49.400","Text":"Let\u0027s not forget the 4."},{"Start":"02:49.400 ","End":"02:55.485","Text":"So 4 times 3 to the x minus 1."},{"Start":"02:55.485 ","End":"02:57.875","Text":"Down below something very similar,"},{"Start":"02:57.875 ","End":"03:06.245","Text":"we just have to do 2x minus x plus 3 and then we\u0027ll get 3 to the x minus 3."},{"Start":"03:06.245 ","End":"03:11.794","Text":"We can also cancel the 3 to the x plus 1 over 3 to the x plus 3."},{"Start":"03:11.794 ","End":"03:14.900","Text":"As before, we just need to subtract the exponents."},{"Start":"03:14.900 ","End":"03:23.205","Text":"That\u0027s this x plus 1 minus x plus 3 is just minus 2,"},{"Start":"03:23.205 ","End":"03:29.450","Text":"so what we get here if we do this division is"},{"Start":"03:29.450 ","End":"03:38.435","Text":"the limit x goes to minus infinity of 3 to the minus 2 is 1/9."},{"Start":"03:38.435 ","End":"03:46.455","Text":"So we get 1/9 and then 4 times 3 to the x"},{"Start":"03:46.455 ","End":"03:55.780","Text":"minus 1 plus 1 over 3 to the x minus 3 plus 1."},{"Start":"03:55.780 ","End":"04:01.990","Text":"At this point, we can just substitute x equals minus infinity, and remember,"},{"Start":"04:01.990 ","End":"04:07.250","Text":"minus infinity minus 1 is still minus infinity and if I subtract 3 from minus infinity,"},{"Start":"04:07.250 ","End":"04:11.840","Text":"it\u0027s still minus infinity and I\u0027m using this thing with a equals 3,"},{"Start":"04:11.840 ","End":"04:15.410","Text":"so this will be 0 and this will be 0."},{"Start":"04:15.410 ","End":"04:21.090","Text":"So what we get is 1/9,"},{"Start":"04:21.090 ","End":"04:30.035","Text":"4 times 0 plus 1 over 0 plus 1."},{"Start":"04:30.035 ","End":"04:33.065","Text":"Well, 4 time 0 plus 1 is just 1,"},{"Start":"04:33.065 ","End":"04:36.605","Text":"and this is also 1 times 1/9,"},{"Start":"04:36.605 ","End":"04:44.110","Text":"so the answer is just 1/9. Well done."}],"ID":4769},{"Watched":false,"Name":"Exercise 15","Duration":"2m 36s","ChapterTopicVideoID":4761,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.840 ","End":"00:09.210","Text":"infinity of the square root of polynomial over a polynomial."},{"Start":"00:09.210 ","End":"00:11.700","Text":"If we just try to substitute infinity,"},{"Start":"00:11.700 ","End":"00:15.960","Text":"we get infinity over infinity here."},{"Start":"00:15.960 ","End":"00:22.335","Text":"Basically, something like square root of infinity over infinity,"},{"Start":"00:22.335 ","End":"00:25.320","Text":"which is undefined, indeterminate."},{"Start":"00:25.320 ","End":"00:27.390","Text":"We have to use another trick,"},{"Start":"00:27.390 ","End":"00:33.960","Text":"and the idea here is that we\u0027re allowed to put the limit inside the square root."},{"Start":"00:33.960 ","End":"00:38.999","Text":"What I mean is that the limit as x goes to infinity of this thing is exactly"},{"Start":"00:38.999 ","End":"00:44.460","Text":"equal to the square root of the limit"},{"Start":"00:44.460 ","End":"00:51.570","Text":"as x goes to infinity of 4x squared"},{"Start":"00:51.570 ","End":"01:00.460","Text":"plus 2 over x squared plus 1000x."},{"Start":"01:02.330 ","End":"01:09.350","Text":"What I\u0027m going to do is just compute the inside without the square root,"},{"Start":"01:09.350 ","End":"01:14.255","Text":"just this part here as a side exercise,"},{"Start":"01:14.255 ","End":"01:17.300","Text":"and then we\u0027ll put it back up here."},{"Start":"01:17.300 ","End":"01:22.580","Text":"What we have is this limit here of the inside,"},{"Start":"01:22.580 ","End":"01:28.385","Text":"that\u0027s the limit as x goes to infinity."},{"Start":"01:28.385 ","End":"01:31.085","Text":"But this kind of exercise we\u0027ve done before,"},{"Start":"01:31.085 ","End":"01:35.360","Text":"we take out the highest power at the top and in the bottom."},{"Start":"01:35.360 ","End":"01:40.430","Text":"It\u0027s x squared in both cases so we take out the x squared."},{"Start":"01:40.430 ","End":"01:48.720","Text":"Here, we\u0027re left with 4 plus 2 over x squared, and here,"},{"Start":"01:48.720 ","End":"01:58.635","Text":"we\u0027re left with x squared times 1 plus 1000 over x."},{"Start":"01:58.635 ","End":"02:01.440","Text":"This thing cancels,"},{"Start":"02:01.440 ","End":"02:05.600","Text":"and we substitute x equals infinity here."},{"Start":"02:05.600 ","End":"02:08.675","Text":"Now, 2 over infinity is 0,"},{"Start":"02:08.675 ","End":"02:11.450","Text":"1000 over infinity is 0,"},{"Start":"02:11.450 ","End":"02:17.120","Text":"so basically, what we get here is just 4."},{"Start":"02:17.120 ","End":"02:21.515","Text":"4 plus 0 over 1 plus 0 is 4."},{"Start":"02:21.515 ","End":"02:24.370","Text":"But now, this was just the inside."},{"Start":"02:24.370 ","End":"02:27.770","Text":"If I put the 4 back into here,"},{"Start":"02:27.770 ","End":"02:31.970","Text":"what I\u0027m going to be left with is the square root of 4,"},{"Start":"02:31.970 ","End":"02:33.725","Text":"which is 2,"},{"Start":"02:33.725 ","End":"02:36.240","Text":"and that\u0027s our answer."}],"ID":4770},{"Watched":false,"Name":"Exercise 16","Duration":"3m 12s","ChapterTopicVideoID":4762,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"In this exercise, we have to find the limit of"},{"Start":"00:03.420 ","End":"00:08.400","Text":"the natural log of this expression, polynomial over polynomial."},{"Start":"00:08.400 ","End":"00:10.890","Text":"Now, if we didn\u0027t have the natural log,"},{"Start":"00:10.890 ","End":"00:12.990","Text":"we\u0027d know how to do this."},{"Start":"00:12.990 ","End":"00:17.700","Text":"The purpose of this exercise is to show that we can actually compute"},{"Start":"00:17.700 ","End":"00:22.870","Text":"the limit of what\u0027s inside the natural log and then apply the natural log."},{"Start":"00:22.870 ","End":"00:24.720","Text":"The first thing to try, of course,"},{"Start":"00:24.720 ","End":"00:27.255","Text":"would be to substitute x equals infinity."},{"Start":"00:27.255 ","End":"00:30.480","Text":"But then we would just get infinity over infinity,"},{"Start":"00:30.480 ","End":"00:34.225","Text":"which is one of those undefined forms."},{"Start":"00:34.225 ","End":"00:42.140","Text":"What we\u0027re going to do is say that this equals the natural logarithm of"},{"Start":"00:42.140 ","End":"00:50.590","Text":"the limit as x goes to infinity of just what\u0027s inside here."},{"Start":"00:50.590 ","End":"00:57.404","Text":"3x cubed minus 5x minus 1"},{"Start":"00:57.404 ","End":"01:05.845","Text":"over x cubed minus 2x squared plus 1."},{"Start":"01:05.845 ","End":"01:13.385","Text":"Now, if I take this and just to aside exercise of computing this part,"},{"Start":"01:13.385 ","End":"01:18.470","Text":"and then I\u0027ll put it back in here and we\u0027ll take the natural log."},{"Start":"01:18.470 ","End":"01:20.690","Text":"This part here,"},{"Start":"01:20.690 ","End":"01:23.220","Text":"we use the usual techniques,"},{"Start":"01:23.220 ","End":"01:26.010","Text":"let\u0027s call it Asterisk."},{"Start":"01:26.010 ","End":"01:30.375","Text":"The asterisk will be the limit."},{"Start":"01:30.375 ","End":"01:34.760","Text":"The usual way to do it is to take out the highest power."},{"Start":"01:34.760 ","End":"01:43.849","Text":"We take out the x cubed and we\u0027re left with 3 minus 5"},{"Start":"01:43.849 ","End":"01:53.010","Text":"over x squared minus 1 over x cubed over,"},{"Start":"01:53.010 ","End":"01:56.300","Text":"and here again, the highest power is x cubed."},{"Start":"01:56.300 ","End":"02:01.820","Text":"It\u0027s x cubed and what we\u0027re left with is 1 minus"},{"Start":"02:01.820 ","End":"02:07.820","Text":"2 over x plus 1 over x cubed."},{"Start":"02:07.820 ","End":"02:10.100","Text":"This cancels."},{"Start":"02:10.100 ","End":"02:19.040","Text":"Since any positive quantity a over infinity is equal to 0."},{"Start":"02:19.040 ","End":"02:22.250","Text":"This is true when a is positive."},{"Start":"02:22.250 ","End":"02:25.100","Text":"If we substitute x equals infinity,"},{"Start":"02:25.100 ","End":"02:27.080","Text":"this is going to be infinity,"},{"Start":"02:27.080 ","End":"02:30.710","Text":"and this is going to be infinity because squared or cubed is still infinity."},{"Start":"02:30.710 ","End":"02:35.860","Text":"In other words, this is going to be 0 and this is 0 and you can see also this and this."},{"Start":"02:35.860 ","End":"02:38.105","Text":"If we substitute now,"},{"Start":"02:38.105 ","End":"02:47.535","Text":"what we get is 3 minus 0 minus 0 over"},{"Start":"02:47.535 ","End":"02:57.435","Text":"1 minus 0 plus 0 and that is just equal to 3."},{"Start":"02:57.435 ","End":"03:00.735","Text":"Now that 3 is the asterisk."},{"Start":"03:00.735 ","End":"03:06.440","Text":"We put that back inside here and we get"},{"Start":"03:06.440 ","End":"03:12.840","Text":"that the answer is the natural log of 3. We\u0027re done."}],"ID":4771},{"Watched":false,"Name":"Exercise 17","Duration":"2m 58s","ChapterTopicVideoID":4763,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:03.990","Text":"Here, we have to find the limit as x goes to"},{"Start":"00:03.990 ","End":"00:08.025","Text":"infinity of e to the power of this whole thing."},{"Start":"00:08.025 ","End":"00:11.370","Text":"This exponent is something that we know how to deal with."},{"Start":"00:11.370 ","End":"00:14.205","Text":"It\u0027s a polynomial over a polynomial."},{"Start":"00:14.205 ","End":"00:18.870","Text":"The purpose of this exercise is to show that we"},{"Start":"00:18.870 ","End":"00:24.320","Text":"can leave the e outside and just figure out the limit of the exponent."},{"Start":"00:24.320 ","End":"00:28.370","Text":"I should first mention that if we just put x equals infinity here,"},{"Start":"00:28.370 ","End":"00:30.710","Text":"we\u0027d get infinity over infinity,"},{"Start":"00:30.710 ","End":"00:33.680","Text":"which is an undefined indeterminate form,"},{"Start":"00:33.680 ","End":"00:36.605","Text":"which is why we need to use this trick."},{"Start":"00:36.605 ","End":"00:47.330","Text":"What we get is that this limit is equal to e to the power of the limit as"},{"Start":"00:47.330 ","End":"00:54.180","Text":"x goes to infinity of x to the 4th plus 2x"},{"Start":"00:54.180 ","End":"01:02.605","Text":"squared plus 6 over 3x to the 4th plus 10x."},{"Start":"01:02.605 ","End":"01:08.855","Text":"In other words, if you take the limit of e to the power of something,"},{"Start":"01:08.855 ","End":"01:13.370","Text":"it\u0027s the same as e to the power of the limit of that same thing."},{"Start":"01:13.370 ","End":"01:16.195","Text":"Now, this part here,"},{"Start":"01:16.195 ","End":"01:18.300","Text":"we\u0027ll do as a side exercise."},{"Start":"01:18.300 ","End":"01:22.085","Text":"Let\u0027s call that an asterisk. Let\u0027s copy it."},{"Start":"01:22.085 ","End":"01:25.850","Text":"That usual technique is to take out the highest power of x."},{"Start":"01:25.850 ","End":"01:29.410","Text":"In this case, it\u0027s x to the 4th,"},{"Start":"01:29.410 ","End":"01:32.645","Text":"and here it\u0027s also going to be x to the 4th."},{"Start":"01:32.645 ","End":"01:38.045","Text":"What we\u0027re left with is 1 plus 2 over x squared,"},{"Start":"01:38.045 ","End":"01:41.455","Text":"6 over x to the 4th,"},{"Start":"01:41.455 ","End":"01:43.875","Text":"and on the denominator,"},{"Start":"01:43.875 ","End":"01:50.805","Text":"3 plus 10 over x cubed."},{"Start":"01:50.805 ","End":"01:55.680","Text":"This cancels with this and at this point,"},{"Start":"01:55.680 ","End":"01:58.860","Text":"we can substitute infinity."},{"Start":"01:58.860 ","End":"02:08.800","Text":"Remember that any number a positive over infinity is equal to 0."},{"Start":"02:08.800 ","End":"02:11.905","Text":"That\u0027s provided that a is positive,"},{"Start":"02:11.905 ","End":"02:15.594","Text":"and so we\u0027re going to get here 0,"},{"Start":"02:15.594 ","End":"02:20.245","Text":"because infinity to the power of 2 or 4 or 3 is going to be infinity."},{"Start":"02:20.245 ","End":"02:23.155","Text":"This, this, and this will be 0."},{"Start":"02:23.155 ","End":"02:28.040","Text":"What we\u0027re left with is 1 plus"},{"Start":"02:28.040 ","End":"02:36.290","Text":"0 plus 0 over 3 plus 0."},{"Start":"02:36.290 ","End":"02:40.480","Text":"Now this is equal to 1/3."},{"Start":"02:40.480 ","End":"02:44.060","Text":"Now remember that\u0027s just the asterisk part."},{"Start":"02:44.060 ","End":"02:50.420","Text":"We have to put this 1/3 back over here and the answer will"},{"Start":"02:50.420 ","End":"02:57.510","Text":"be e to the power of 1/3 and we\u0027re done."}],"ID":4772},{"Watched":false,"Name":"Exercise 18","Duration":"3m 44s","ChapterTopicVideoID":4764,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.320","Text":"In this exercise, we have to figure out the limit as x goes to minus infinity of"},{"Start":"00:05.320 ","End":"00:11.275","Text":"the sign of this rational function polynomial over polynomial."},{"Start":"00:11.275 ","End":"00:16.580","Text":"We first try naively to substitute minus infinity."},{"Start":"00:16.580 ","End":"00:20.965","Text":"If I put minus infinity in the numerator here,"},{"Start":"00:20.965 ","End":"00:23.110","Text":"it behaves like the highest power,"},{"Start":"00:23.110 ","End":"00:24.370","Text":"so it\u0027s like x^4,"},{"Start":"00:24.370 ","End":"00:25.855","Text":"it goes to infinity."},{"Start":"00:25.855 ","End":"00:28.840","Text":"The denominator behaves like minus infinity."},{"Start":"00:28.840 ","End":"00:31.195","Text":"We have one of those infinity over infinity."},{"Start":"00:31.195 ","End":"00:33.460","Text":"Can\u0027t do it by substitution."},{"Start":"00:33.460 ","End":"00:36.710","Text":"We\u0027re going to have to use some tricks."},{"Start":"00:36.710 ","End":"00:40.374","Text":"The usual one is when you have a sign,"},{"Start":"00:40.374 ","End":"00:45.119","Text":"is to exchange the limit with the sign."},{"Start":"00:45.119 ","End":"00:47.735","Text":"What we going to do first is,"},{"Start":"00:47.735 ","End":"00:51.635","Text":"say that this is the sign of"},{"Start":"00:51.635 ","End":"00:58.010","Text":"the limit as x goes to minus infinity of same thing,"},{"Start":"00:58.010 ","End":"01:02.360","Text":"just copy it, x^ 4 plus 2x squared plus 6"},{"Start":"01:02.360 ","End":"01:08.305","Text":"over 3x^5 plus 10x."},{"Start":"01:08.305 ","End":"01:12.490","Text":"That would do the limit of this and then AN will do the sign of what we get."},{"Start":"01:12.490 ","End":"01:16.715","Text":"What I\u0027d like to do is just do this bit as side exercise,"},{"Start":"01:16.715 ","End":"01:18.365","Text":"let\u0027s say I\u0027ll do it over here,"},{"Start":"01:18.365 ","End":"01:22.550","Text":"call it and I don\u0027t know asterisk and do the asterisk over here."},{"Start":"01:22.550 ","End":"01:27.955","Text":"What I\u0027ve got is the limit as x goes to minus infinity."},{"Start":"01:27.955 ","End":"01:33.170","Text":"I\u0027m going to use a standard trick with polynomials over polynomials."},{"Start":"01:33.170 ","End":"01:37.670","Text":"We take out the highest power in each of the numerator and denominator separately."},{"Start":"01:37.670 ","End":"01:40.705","Text":"In here, x^4 is the highest power."},{"Start":"01:40.705 ","End":"01:42.990","Text":"I\u0027ll take it outside the brackets in a moment."},{"Start":"01:42.990 ","End":"01:45.620","Text":"In here, it\u0027ll be x^5."},{"Start":"01:45.620 ","End":"01:47.075","Text":"Let\u0027s see what we\u0027re left with."},{"Start":"01:47.075 ","End":"01:51.750","Text":"Here we have 1 plus dividing by x^4,"},{"Start":"01:51.750 ","End":"01:54.345","Text":"that becomes 2 over x squared,"},{"Start":"01:54.345 ","End":"01:57.735","Text":"and here, 6 over x^4."},{"Start":"01:57.735 ","End":"02:00.120","Text":"Here, if I take x^5 out,"},{"Start":"02:00.120 ","End":"02:02.025","Text":"I\u0027m left with 3 here."},{"Start":"02:02.025 ","End":"02:04.905","Text":"X over x^5 is x^4,"},{"Start":"02:04.905 ","End":"02:07.890","Text":"so it\u0027s 10 over x^4."},{"Start":"02:07.890 ","End":"02:12.470","Text":"Now I can do the limit as x goes to minus infinity."},{"Start":"02:12.470 ","End":"02:14.245","Text":"First of all, note,"},{"Start":"02:14.245 ","End":"02:17.820","Text":"that I can cancel x^4 over x^5"},{"Start":"02:17.820 ","End":"02:20.460","Text":"just gives me 1 over x."},{"Start":"02:20.460 ","End":"02:23.115","Text":"I just have an x in the bottom."},{"Start":"02:23.115 ","End":"02:27.270","Text":"What I get is,"},{"Start":"02:27.270 ","End":"02:28.470","Text":"I can now do the limit."},{"Start":"02:28.470 ","End":"02:35.910","Text":"The 1 over x is like 1 over infinity minus infinity."},{"Start":"02:37.520 ","End":"02:40.470","Text":"I don\u0027t need the 1."},{"Start":"02:40.470 ","End":"02:44.505","Text":"What I have here is 1 plus,"},{"Start":"02:44.505 ","End":"02:47.780","Text":"now 2 over infinity squared or"},{"Start":"02:47.780 ","End":"02:51.590","Text":"minus infinity squared doesn\u0027t even matter, is going to be 0."},{"Start":"02:51.590 ","End":"02:54.320","Text":"A number over infinity is 0."},{"Start":"02:54.320 ","End":"02:58.310","Text":"Here also, we are going to get 0, 6 times 0."},{"Start":"02:58.310 ","End":"03:05.010","Text":"Here we\u0027re going to get 3 plus 0. Now here\u0027s no problem."},{"Start":"03:05.010 ","End":"03:06.650","Text":"This is 1, this is 3,"},{"Start":"03:06.650 ","End":"03:07.985","Text":"but here we\u0027re having,"},{"Start":"03:07.985 ","End":"03:09.350","Text":"I\u0027ll leave the 1 in again."},{"Start":"03:09.350 ","End":"03:15.095","Text":"Anyway, 1 over minus infinity is minus 1 over infinity is 0."},{"Start":"03:15.095 ","End":"03:20.180","Text":"I\u0027ll just emphasize that this is the important thing and this is 0"},{"Start":"03:20.180 ","End":"03:26.645","Text":"so what we get is 0 times 1 over 3, It\u0027s just 0."},{"Start":"03:26.645 ","End":"03:30.580","Text":"Now I\u0027m going to go back here and put the 0 in,"},{"Start":"03:30.580 ","End":"03:33.410","Text":"and so this equals to sine,"},{"Start":"03:33.410 ","End":"03:36.365","Text":"and this whole thing is replaced by 0,"},{"Start":"03:36.365 ","End":"03:39.335","Text":"and sine of 0 is 0."},{"Start":"03:39.335 ","End":"03:43.950","Text":"This is the answer to the question. We\u0027re done."}],"ID":4773},{"Watched":false,"Name":"Exercise 19","Duration":"4m 37s","ChapterTopicVideoID":4765,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.315","Text":"Here we have to find the limit as x goes to infinity of this expression."},{"Start":"00:06.315 ","End":"00:10.620","Text":"If you notice, it\u0027s exactly the same as the previous exercise"},{"Start":"00:10.620 ","End":"00:16.200","Text":"where k was equal to 5 and here it\u0027s just more general."},{"Start":"00:16.200 ","End":"00:20.600","Text":"I\u0027m going to go through this a bit quicker because we\u0027ve done it before."},{"Start":"00:20.600 ","End":"00:23.730","Text":"Trying to substitute x equals infinity doesn\u0027t work,"},{"Start":"00:23.730 ","End":"00:26.585","Text":"it just gives us infinity minus infinity,"},{"Start":"00:26.585 ","End":"00:29.360","Text":"which is undefined, could be anything."},{"Start":"00:29.360 ","End":"00:34.025","Text":"We\u0027re going to use some algebra and we\u0027re going to use conjugates."},{"Start":"00:34.025 ","End":"00:38.330","Text":"That I will remind you again that the conjugate,"},{"Start":"00:38.330 ","End":"00:42.964","Text":"if we have something of the form a minus b,"},{"Start":"00:42.964 ","End":"00:48.590","Text":"then its conjugate is a plus b and vice versa,"},{"Start":"00:48.590 ","End":"00:51.695","Text":"these 2 expressions are conjugates of each other,"},{"Start":"00:51.695 ","End":"00:55.020","Text":"and the useful thing is that if you multiply them,"},{"Start":"00:55.020 ","End":"00:58.080","Text":"a minus b times a plus b,"},{"Start":"00:58.080 ","End":"01:03.190","Text":"is a difference of squares formula and it comes out to be a squared minus b squared,"},{"Start":"01:03.190 ","End":"01:09.360","Text":"so that if a or b is the square root after squaring no longer is."},{"Start":"01:10.090 ","End":"01:16.565","Text":"The other thing that\u0027s going to come in handy for later on is that if you have a number"},{"Start":"01:16.565 ","End":"01:22.820","Text":"a and you divide it by infinity could be plus or minus infinity, that equals 0."},{"Start":"01:22.820 ","End":"01:24.920","Text":"That might also be useful."},{"Start":"01:24.920 ","End":"01:26.945","Text":"What we\u0027ll do is,"},{"Start":"01:26.945 ","End":"01:34.160","Text":"take the limit as x goes to infinity and multiply by the conjugate,"},{"Start":"01:34.160 ","End":"01:39.065","Text":"so we get the square root of"},{"Start":"01:39.065 ","End":"01:44.610","Text":"x squared plus kx minus x,"},{"Start":"01:44.610 ","End":"01:47.820","Text":"all this times the conjugate."},{"Start":"01:47.820 ","End":"01:55.429","Text":"Square root of x squared plus kx plus x,"},{"Start":"01:55.429 ","End":"01:57.770","Text":"just change the sign from minus to a plus."},{"Start":"01:57.770 ","End":"01:59.840","Text":"But I can\u0027t just multiply by something,"},{"Start":"01:59.840 ","End":"02:03.080","Text":"I changed the exercise, I\u0027ve to compensate."},{"Start":"02:03.080 ","End":"02:12.970","Text":"If I divide, I multiplied by the square root of x squared plus kx plus x."},{"Start":"02:12.970 ","End":"02:16.790","Text":"If I cancel these 2 or multiplying by this over this which is 1,"},{"Start":"02:16.790 ","End":"02:19.640","Text":"so now I haven\u0027t changed the exercise."},{"Start":"02:19.640 ","End":"02:26.360","Text":"If I do the multiplication I just get this thing squared,"},{"Start":"02:26.360 ","End":"02:29.390","Text":"which is x squared plus kx minus x squared,"},{"Start":"02:29.390 ","End":"02:33.050","Text":"I\u0027m left in the numerator with just kx,"},{"Start":"02:33.050 ","End":"02:35.420","Text":"and then in the denominator,"},{"Start":"02:35.420 ","End":"02:41.450","Text":"we have the square root of x squared times"},{"Start":"02:41.450 ","End":"02:48.860","Text":"1 plus k over x, plus x."},{"Start":"02:48.860 ","End":"02:53.030","Text":"Now this equals the square root is,"},{"Start":"02:53.030 ","End":"02:56.000","Text":"by the rules of square root of a product is"},{"Start":"02:56.000 ","End":"03:00.005","Text":"just the product of the square roots provided that they\u0027re all positive."},{"Start":"03:00.005 ","End":"03:02.225","Text":"This is positive, of course,"},{"Start":"03:02.225 ","End":"03:10.030","Text":"and 1 plus k over x is also positive because k over x is pretty small."},{"Start":"03:10.040 ","End":"03:13.460","Text":"Even if k is negative, actually is big enough,"},{"Start":"03:13.460 ","End":"03:16.625","Text":"this thing is positive, so that\u0027s okay."},{"Start":"03:16.625 ","End":"03:22.110","Text":"The square root of x squared is the absolute value of x,"},{"Start":"03:22.110 ","End":"03:24.475","Text":"but when x is positive,"},{"Start":"03:24.475 ","End":"03:28.459","Text":"then the absolute value of x is just x itself."},{"Start":"03:28.459 ","End":"03:33.319","Text":"We end up getting this thing is x times"},{"Start":"03:33.319 ","End":"03:39.585","Text":"the square root of 1 plus k over x,"},{"Start":"03:39.585 ","End":"03:42.170","Text":"and then the x comes out of the brackets."},{"Start":"03:42.170 ","End":"03:47.595","Text":"Basically what I can do is cancel x with x and with x,"},{"Start":"03:47.595 ","End":"03:49.700","Text":"because x comes out of the brackets and cancels."},{"Start":"03:49.700 ","End":"03:56.630","Text":"What we\u0027re left with is the limit as x goes to infinity of"},{"Start":"03:56.630 ","End":"04:01.460","Text":"k over the square root"},{"Start":"04:01.460 ","End":"04:07.620","Text":"of 1 plus k over x plus 1."},{"Start":"04:07.620 ","End":"04:12.920","Text":"Now, what I mentioned here about a over infinity being 0 at this point,"},{"Start":"04:12.920 ","End":"04:15.200","Text":"our substitute x equals infinity,"},{"Start":"04:15.200 ","End":"04:17.090","Text":"and this part here,"},{"Start":"04:17.090 ","End":"04:20.045","Text":"this just becomes what I\u0027ve circled."},{"Start":"04:20.045 ","End":"04:23.135","Text":"I get square root of 1 is 1 plus 1 is 2,"},{"Start":"04:23.135 ","End":"04:28.830","Text":"so the answer would just be K over 2."},{"Start":"04:28.830 ","End":"04:30.525","Text":"That\u0027s our answer."},{"Start":"04:30.525 ","End":"04:32.990","Text":"I just got to show you that infinity minus"},{"Start":"04:32.990 ","End":"04:36.870","Text":"infinity can be practically anything. We\u0027re done."}],"ID":4774},{"Watched":false,"Name":"Exercise 20","Duration":"3m 29s","ChapterTopicVideoID":4766,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"Here we have to find the limit as x goes to infinity of this expression."},{"Start":"00:04.830 ","End":"00:07.935","Text":"We\u0027ve seen this thing many times before."},{"Start":"00:07.935 ","End":"00:11.430","Text":"A straight substitution gives infinity minus infinity,"},{"Start":"00:11.430 ","End":"00:15.330","Text":"and so we multiply and divide by the conjugate."},{"Start":"00:15.330 ","End":"00:23.400","Text":"What we get is that this is equal to the limit as x goes to"},{"Start":"00:23.400 ","End":"00:28.500","Text":"infinity of the square root of"},{"Start":"00:28.500 ","End":"00:34.260","Text":"x squared plus x plus 1 minus x,"},{"Start":"00:34.260 ","End":"00:38.775","Text":"and I\u0027m going to multiply this by its conjugate,"},{"Start":"00:38.775 ","End":"00:44.420","Text":"square root of x squared plus x plus 1 conjugate,"},{"Start":"00:44.420 ","End":"00:48.350","Text":"meaning we put plus x instead of minus x here."},{"Start":"00:48.350 ","End":"00:52.180","Text":"But we can\u0027t just multiply by something without dividing it,"},{"Start":"00:52.180 ","End":"00:54.665","Text":"so we also divide by the same thing."},{"Start":"00:54.665 ","End":"01:03.395","Text":"Square root of x squared plus x plus 1 plus x."},{"Start":"01:03.395 ","End":"01:09.920","Text":"This equals limit of this thing squared"},{"Start":"01:09.920 ","End":"01:18.524","Text":"is x squared plus x plus 1 minus this thing squared."},{"Start":"01:18.524 ","End":"01:21.495","Text":"Just as a quick reminder,"},{"Start":"01:21.495 ","End":"01:26.610","Text":"A minus B times its conjugate A plus B"},{"Start":"01:26.610 ","End":"01:30.155","Text":"is equal to A squared minus B squared."},{"Start":"01:30.155 ","End":"01:32.644","Text":"Famous formula from algebra."},{"Start":"01:32.644 ","End":"01:35.440","Text":"That\u0027s what I\u0027ve done in the numerator."},{"Start":"01:35.440 ","End":"01:37.225","Text":"On the denominator,"},{"Start":"01:37.225 ","End":"01:40.865","Text":"I\u0027m going to take x squared outside the brackets."},{"Start":"01:40.865 ","End":"01:43.670","Text":"Since we\u0027ve done this so many times before,"},{"Start":"01:43.670 ","End":"01:45.230","Text":"I\u0027ll cut out a step."},{"Start":"01:45.230 ","End":"01:50.655","Text":"So square root of x squared times the square root of,"},{"Start":"01:50.655 ","End":"01:52.580","Text":"when I take x squared out,"},{"Start":"01:52.580 ","End":"02:01.170","Text":"I\u0027m left with 1 plus 1 over x plus 1 over x squared plus x."},{"Start":"02:01.170 ","End":"02:04.900","Text":"The square root of x squared is equal"},{"Start":"02:04.900 ","End":"02:07.225","Text":"to the absolute value of x in general,"},{"Start":"02:07.225 ","End":"02:11.780","Text":"but is just equal to x when x is positive."},{"Start":"02:11.780 ","End":"02:14.844","Text":"Which it is because it\u0027s going to plus infinity."},{"Start":"02:14.844 ","End":"02:21.850","Text":"This is x equals limit x tends to infinity."},{"Start":"02:21.850 ","End":"02:24.519","Text":"I\u0027ll take x out of the numerator."},{"Start":"02:24.519 ","End":"02:28.670","Text":"Also, notice that this and this cancels."},{"Start":"02:28.670 ","End":"02:36.635","Text":"If I take x, I just get 1 plus 1 over x divided by,"},{"Start":"02:36.635 ","End":"02:40.535","Text":"now this thing, square root of x squared is just x."},{"Start":"02:40.535 ","End":"02:43.945","Text":"What we\u0027re left is if we take it outside the brackets,"},{"Start":"02:43.945 ","End":"02:47.900","Text":"is this square root here plus 1."},{"Start":"02:47.900 ","End":"02:49.445","Text":"I\u0027ll just copy what\u0027s here."},{"Start":"02:49.445 ","End":"02:55.055","Text":"1 plus 1 over x plus 1 over x squared."},{"Start":"02:55.055 ","End":"02:58.780","Text":"This x will cancel with this x,"},{"Start":"02:58.780 ","End":"03:04.190","Text":"and if we use the formula that any number"},{"Start":"03:04.190 ","End":"03:09.050","Text":"over infinity is equal to 0,"},{"Start":"03:09.050 ","End":"03:13.045","Text":"and here we\u0027ll get a 0, we get a 0 here and here."},{"Start":"03:13.045 ","End":"03:17.915","Text":"What we\u0027re going to be left with is a numerator 1,"},{"Start":"03:17.915 ","End":"03:19.565","Text":"and in the denominator,"},{"Start":"03:19.565 ","End":"03:22.220","Text":"square root of 1 plus 1 is 2,"},{"Start":"03:22.220 ","End":"03:25.640","Text":"so 1 over 2,"},{"Start":"03:25.640 ","End":"03:29.110","Text":"1/2 as the answer. We\u0027re done."}],"ID":4775},{"Watched":false,"Name":"Exercise 21","Duration":"5m 41s","ChapterTopicVideoID":4767,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"This exercise is very similar to the previous one,"},{"Start":"00:04.725 ","End":"00:06.480","Text":"where we had a minus x here,"},{"Start":"00:06.480 ","End":"00:08.685","Text":"and we had a plus infinity here."},{"Start":"00:08.685 ","End":"00:12.525","Text":"Here we have to find the limit as x goes to minus infinity of this."},{"Start":"00:12.525 ","End":"00:15.430","Text":"If we substitute minus infinity,"},{"Start":"00:15.430 ","End":"00:20.890","Text":"you\u0027re going to get into trouble because minus infinity squared is infinity."},{"Start":"00:20.890 ","End":"00:22.525","Text":"Here we have minus infinity."},{"Start":"00:22.525 ","End":"00:25.010","Text":"Already we have an infinity minus infinity"},{"Start":"00:25.010 ","End":"00:29.160","Text":"and no purpose in continuing to substitute anymore."},{"Start":"00:29.160 ","End":"00:33.170","Text":"We\u0027ll use the method of the conjugates as we did before."},{"Start":"00:33.170 ","End":"00:39.500","Text":"Just a quick reminder that the conjugate of A plus B,"},{"Start":"00:39.500 ","End":"00:45.180","Text":"this will be A, this will be B is A minus B, and vice versa."},{"Start":"00:45.180 ","End":"00:47.460","Text":"The conjugate of A minus B is a plus B."},{"Start":"00:47.460 ","End":"00:54.274","Text":"The advantage of using conjugates is that if you multiply an expression by its conjugate,"},{"Start":"00:54.274 ","End":"00:57.485","Text":"you get what is called the difference of squares formula."},{"Start":"00:57.485 ","End":"01:01.210","Text":"If either A or B is the square root after squaring it no longer is."},{"Start":"01:01.210 ","End":"01:05.060","Text":"The other thing that we\u0027re going to need is that any number"},{"Start":"01:05.060 ","End":"01:11.120","Text":"over infinity could be plus infinity or minus infinity is 0."},{"Start":"01:11.120 ","End":"01:20.000","Text":"Rewriting this thing, this equals the limit as x tends to"},{"Start":"01:20.000 ","End":"01:25.970","Text":"minus infinity of the square root of"},{"Start":"01:25.970 ","End":"01:32.970","Text":"x squared plus x plus 1 plus x."},{"Start":"01:32.970 ","End":"01:35.869","Text":"Since I can\u0027t just multiply by the conjugate,"},{"Start":"01:35.869 ","End":"01:37.220","Text":"I\u0027m going to have to compensate."},{"Start":"01:37.220 ","End":"01:39.800","Text":"But if I put the same thing on the top, and on the bottom,"},{"Start":"01:39.800 ","End":"01:45.395","Text":"I\u0027m multiplying by 1 times square root of"},{"Start":"01:45.395 ","End":"01:51.605","Text":"x squared plus x plus 1 this time minus x."},{"Start":"01:51.605 ","End":"01:53.930","Text":"The same thing on the denominator,"},{"Start":"01:53.930 ","End":"02:02.570","Text":"square root of x squared plus x plus 1 minus x,"},{"Start":"02:02.570 ","End":"02:09.785","Text":"which equals limit x goes to minus infinity."},{"Start":"02:09.785 ","End":"02:13.985","Text":"Multiplying this by this using the difference of squares formula,"},{"Start":"02:13.985 ","End":"02:22.320","Text":"we get x squared plus x plus 1 minus x squared."},{"Start":"02:22.320 ","End":"02:28.290","Text":"This thing cancels over the square root."},{"Start":"02:28.290 ","End":"02:31.150","Text":"We\u0027re going to use the same trick as we usually use."},{"Start":"02:31.150 ","End":"02:34.540","Text":"Take x squared outside the brackets under the square root and"},{"Start":"02:34.540 ","End":"02:40.165","Text":"then factor it as square root of x squared separately."},{"Start":"02:40.165 ","End":"02:41.770","Text":"The square root is"},{"Start":"02:41.770 ","End":"02:51.835","Text":"1 plus 1 over x plus 1 over x squared minus x."},{"Start":"02:51.835 ","End":"02:55.645","Text":"Then we take x outside the brackets,"},{"Start":"02:55.645 ","End":"03:02.270","Text":"and we\u0027re left with limit x goes to minus infinity."},{"Start":"03:02.270 ","End":"03:04.780","Text":"If I take x out of this,"},{"Start":"03:04.780 ","End":"03:11.620","Text":"I get 1 plus 1 over x divided by,"},{"Start":"03:11.620 ","End":"03:13.750","Text":"what is the square root of x squared?"},{"Start":"03:13.750 ","End":"03:20.350","Text":"The square root of x squared is the absolute value of x."},{"Start":"03:20.350 ","End":"03:23.485","Text":"But when x is negative,"},{"Start":"03:23.485 ","End":"03:26.860","Text":"even very negative, it\u0027s going to minus infinity."},{"Start":"03:26.860 ","End":"03:27.940","Text":"When x is negative,"},{"Start":"03:27.940 ","End":"03:31.450","Text":"the absolute value of x is minus x."},{"Start":"03:31.450 ","End":"03:35.485","Text":"For example, the square root of, say,"},{"Start":"03:35.485 ","End":"03:39.605","Text":"a million, which would be minus 1,000 squared,"},{"Start":"03:39.605 ","End":"03:41.640","Text":"would be plus 1,000."},{"Start":"03:41.640 ","End":"03:43.395","Text":"For x very negative,"},{"Start":"03:43.395 ","End":"03:45.810","Text":"we need to get something positive for the square roots."},{"Start":"03:45.810 ","End":"03:47.020","Text":"There\u0027s a minus here,"},{"Start":"03:47.020 ","End":"03:49.175","Text":"that\u0027s what you have to watch out for."},{"Start":"03:49.175 ","End":"03:55.510","Text":"Here we have minus x times same square root,"},{"Start":"03:55.510 ","End":"04:03.060","Text":"1 plus 1 over x plus 1 over x squared minus x."},{"Start":"04:03.060 ","End":"04:06.155","Text":"We\u0027ve done this thing before."},{"Start":"04:06.155 ","End":"04:09.830","Text":"I\u0027m going to take x out of the bottom also,"},{"Start":"04:09.830 ","End":"04:11.975","Text":"and then cancel the xs."},{"Start":"04:11.975 ","End":"04:18.800","Text":"I cancel the x from here together with the x from here and the x from here,"},{"Start":"04:18.800 ","End":"04:21.220","Text":"leaving it just with 1."},{"Start":"04:21.220 ","End":"04:27.875","Text":"Because I took x outside the brackets that get minus a square root minus 1."},{"Start":"04:27.875 ","End":"04:34.820","Text":"We get 1 plus 1 over x on the numerator."},{"Start":"04:34.820 ","End":"04:37.070","Text":"Now, the denominator,"},{"Start":"04:37.070 ","End":"04:43.055","Text":"square root of 1 plus 1 over x"},{"Start":"04:43.055 ","End":"04:48.260","Text":"plus 1 over x squared minus 1."},{"Start":"04:48.260 ","End":"04:51.980","Text":"Now I\u0027m going to use what I wrote here."},{"Start":"04:51.980 ","End":"04:56.045","Text":"Something over plus or minus infinity is 0."},{"Start":"04:56.045 ","End":"04:59.965","Text":"Here we have 1 over infinity and that\u0027s 0."},{"Start":"04:59.965 ","End":"05:03.560","Text":"At this point, we\u0027re going to actually substitute minus infinity."},{"Start":"05:03.560 ","End":"05:10.355","Text":"We get 1 plus 0 over"},{"Start":"05:10.355 ","End":"05:18.860","Text":"the square root of 1 plus 0 plus 0 minus 1."},{"Start":"05:18.860 ","End":"05:20.720","Text":"I forgot the minus here."},{"Start":"05:20.720 ","End":"05:22.580","Text":"This is minus here."},{"Start":"05:22.580 ","End":"05:28.535","Text":"Minus 1, could see something was going to be wrong if I did 1 minus 1 here."},{"Start":"05:28.535 ","End":"05:30.335","Text":"This is just 1."},{"Start":"05:30.335 ","End":"05:32.540","Text":"Here we have a square root of 1 is 1,"},{"Start":"05:32.540 ","End":"05:34.475","Text":"1 over minus 2,"},{"Start":"05:34.475 ","End":"05:38.340","Text":"in other words, minus a half."},{"Start":"05:38.340 ","End":"05:40.480","Text":"That\u0027s the answer."}],"ID":4776},{"Watched":false,"Name":"Exercise 22","Duration":"5m 33s","ChapterTopicVideoID":4768,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:02.340","Text":"In this exercise,"},{"Start":"00:02.340 ","End":"00:07.245","Text":"we have to find the limit as x goes to infinity of this expression."},{"Start":"00:07.245 ","End":"00:10.920","Text":"We\u0027ve done many of this kind before that if you just put infinity in here,"},{"Start":"00:10.920 ","End":"00:12.660","Text":"you get the square root of infinity,"},{"Start":"00:12.660 ","End":"00:17.100","Text":"which is infinity minus infinity and infinity minus"},{"Start":"00:17.100 ","End":"00:22.950","Text":"infinity is 1 of those undefined things and the answer could be anything."},{"Start":"00:22.950 ","End":"00:25.875","Text":"We use our usual tricks here."},{"Start":"00:25.875 ","End":"00:30.030","Text":"It\u0027s clear that we should use the conjugate because we have a square root here."},{"Start":"00:30.030 ","End":"00:33.450","Text":"I\u0027ll just give you a quick reminder of what a conjugate is."},{"Start":"00:33.450 ","End":"00:37.995","Text":"If we have an expression of the form A minus B,"},{"Start":"00:37.995 ","End":"00:44.340","Text":"then its conjugate is just change the sign or the other way around."},{"Start":"00:44.340 ","End":"00:48.330","Text":"These are conjugate to each other and the advantage of using this is that,"},{"Start":"00:48.330 ","End":"00:53.310","Text":"if we multiply A minus B by A plus B,"},{"Start":"00:53.310 ","End":"00:54.915","Text":"two conjugates together,"},{"Start":"00:54.915 ","End":"00:57.990","Text":"we get a difference of squares formula,"},{"Start":"00:57.990 ","End":"00:59.820","Text":"A squared minus B squared,"},{"Start":"00:59.820 ","End":"01:01.650","Text":"so if one of these is a square root,"},{"Start":"01:01.650 ","End":"01:05.340","Text":"when you square it, it will no longer be a square root."},{"Start":"01:05.340 ","End":"01:07.530","Text":"I\u0027ll just remind you that,"},{"Start":"01:07.530 ","End":"01:11.235","Text":"if I have a number A and I divide it by infinity,"},{"Start":"01:11.235 ","End":"01:13.230","Text":"could be plus or minus,"},{"Start":"01:13.230 ","End":"01:15.405","Text":"then that will be 0."},{"Start":"01:15.405 ","End":"01:23.370","Text":"So, the limit as x goes to infinity of"},{"Start":"01:23.370 ","End":"01:32.805","Text":"this square root of x to the fourth plus x squared plus 1 minus x squared."},{"Start":"01:32.805 ","End":"01:37.200","Text":"What I\u0027m going to do to make it over a fraction and multiply top and"},{"Start":"01:37.200 ","End":"01:41.760","Text":"bottom by the conjugate and if I multiply top and bottom by the same quantity,"},{"Start":"01:41.760 ","End":"01:43.530","Text":"it\u0027s like multiplying by 1,"},{"Start":"01:43.530 ","End":"01:45.900","Text":"so I haven\u0027t changed the exercise."},{"Start":"01:45.900 ","End":"01:49.860","Text":"Right here, the same square root,"},{"Start":"01:49.860 ","End":"01:53.670","Text":"only this time it will be plus x squared."},{"Start":"01:53.670 ","End":"01:56.370","Text":"Just copy it, x to the fourth,"},{"Start":"01:56.370 ","End":"02:03.435","Text":"plus x squared plus 1 and the difference is here\u0027s minus here\u0027s plus."},{"Start":"02:03.435 ","End":"02:06.420","Text":"But we have to write the same thing on the denominator,"},{"Start":"02:06.420 ","End":"02:08.175","Text":"so here also,"},{"Start":"02:08.175 ","End":"02:12.885","Text":"we have the square root of x to the fourth,"},{"Start":"02:12.885 ","End":"02:18.960","Text":"plus x squared plus 1 plus x squared,"},{"Start":"02:18.960 ","End":"02:20.760","Text":"so this thing is the same as this thing,"},{"Start":"02:20.760 ","End":"02:23.310","Text":"so we haven\u0027t changed the value."},{"Start":"02:23.310 ","End":"02:30.585","Text":"Now, limit as x goes to infinity,"},{"Start":"02:30.585 ","End":"02:34.215","Text":"this thing squared is just as it is."},{"Start":"02:34.215 ","End":"02:35.895","Text":"X to the fourth,"},{"Start":"02:35.895 ","End":"02:39.165","Text":"plus x squared plus 1."},{"Start":"02:39.165 ","End":"02:42.510","Text":"This thing squared minus x squared squared,"},{"Start":"02:42.510 ","End":"02:45.540","Text":"which is minus x to the fourth."},{"Start":"02:45.540 ","End":"02:50.550","Text":"That\u0027s the numerator and the denominator is what it is here,"},{"Start":"02:50.550 ","End":"02:53.880","Text":"but let\u0027s do some simplification. We\u0027ve done this before."},{"Start":"02:53.880 ","End":"03:00.510","Text":"We take x to the fourth outside the brackets and we get the square root and then we use"},{"Start":"03:00.510 ","End":"03:07.665","Text":"the formula that the square root of ab is square root of a square root of b,"},{"Start":"03:07.665 ","End":"03:10.455","Text":"and if we do this with a equals x to the fourth,"},{"Start":"03:10.455 ","End":"03:15.720","Text":"x to the fourth times 1 plus 1"},{"Start":"03:15.720 ","End":"03:21.045","Text":"over x squared plus 1 over x to the fourth,"},{"Start":"03:21.045 ","End":"03:25.095","Text":"close the bracket plus x squared."},{"Start":"03:25.095 ","End":"03:26.985","Text":"Now, we\u0027re lucky,"},{"Start":"03:26.985 ","End":"03:32.925","Text":"this thing cancels out and what we\u0027re going to do now is take"},{"Start":"03:32.925 ","End":"03:39.450","Text":"x squared outside the brackets here and that way it will cancel with this x squared,"},{"Start":"03:39.450 ","End":"03:45.225","Text":"so we get limit x goes to infinity"},{"Start":"03:45.225 ","End":"03:53.790","Text":"of x squared brackets 1 plus 1 over x squared."},{"Start":"03:53.790 ","End":"03:59.325","Text":"Here, notice that we have the square root of x to the fourth."},{"Start":"03:59.325 ","End":"04:02.715","Text":"This would normally be plus or minus x squared,"},{"Start":"04:02.715 ","End":"04:05.370","Text":"but because x is going to infinity,"},{"Start":"04:05.370 ","End":"04:08.850","Text":"then it\u0027s going to be plus x squared."},{"Start":"04:08.850 ","End":"04:11.760","Text":"The square root of x to the fourth is always x squared because"},{"Start":"04:11.760 ","End":"04:15.270","Text":"the square root has to be positive and x squared is positive,"},{"Start":"04:15.270 ","End":"04:16.905","Text":"so it has to be x squared,"},{"Start":"04:16.905 ","End":"04:19.330","Text":"can\u0027t be minus x squared."},{"Start":"04:21.140 ","End":"04:23.895","Text":"In the denominator,"},{"Start":"04:23.895 ","End":"04:27.930","Text":"this is x squared times same thing,"},{"Start":"04:27.930 ","End":"04:35.310","Text":"1 plus 1 over x squared plus 1 over x to the fourth plus x squared."},{"Start":"04:35.310 ","End":"04:38.355","Text":"If we take x squared out of the brackets here,"},{"Start":"04:38.355 ","End":"04:43.635","Text":"we\u0027ll get this thing plus 1 until the x squared basically cancel."},{"Start":"04:43.635 ","End":"04:47.250","Text":"What I\u0027m saying, is we can skip a step because we\u0027re good at algebra,"},{"Start":"04:47.250 ","End":"04:49.170","Text":"and this cancels with this,"},{"Start":"04:49.170 ","End":"04:50.250","Text":"this cancels with this,"},{"Start":"04:50.250 ","End":"04:53.080","Text":"but leaves a 1 here."},{"Start":"04:53.750 ","End":"04:58.770","Text":"What we have now is that we can now substitute x equals infinity,"},{"Start":"04:58.770 ","End":"05:00.885","Text":"and so we get, from here,"},{"Start":"05:00.885 ","End":"05:04.035","Text":"we get a 1, from here,"},{"Start":"05:04.035 ","End":"05:11.205","Text":"we get a 0 because infinity 1 over infinity is 0"},{"Start":"05:11.205 ","End":"05:21.270","Text":"divided by 1 plus 0 plus 0 plus 1."},{"Start":"05:21.270 ","End":"05:26.970","Text":"This is 1, this is 1 plus 0 plus 1 is 2,"},{"Start":"05:26.970 ","End":"05:32.860","Text":"so it\u0027s just equal to 1/2 and that\u0027s the answer."}],"ID":4777},{"Watched":false,"Name":"Exercise 23","Duration":"5m 32s","ChapterTopicVideoID":4769,"CourseChapterTopicPlaylistID":3700,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.745","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression,"},{"Start":"00:05.745 ","End":"00:08.535","Text":"the square root of this minus the square root of that."},{"Start":"00:08.535 ","End":"00:11.640","Text":"We\u0027re pretty familiar with this exercise already."},{"Start":"00:11.640 ","End":"00:14.550","Text":"If we substitute x equals infinity,"},{"Start":"00:14.550 ","End":"00:18.195","Text":"we\u0027re going to get the square root of infinity minus the square root infinity."},{"Start":"00:18.195 ","End":"00:23.355","Text":"In other words, we\u0027re going to get the expression of the type infinity minus infinity."},{"Start":"00:23.355 ","End":"00:26.385","Text":"We\u0027ve already seen could be anything, it\u0027s not defined."},{"Start":"00:26.385 ","End":"00:28.620","Text":"We have to use other techniques."},{"Start":"00:28.620 ","End":"00:30.690","Text":"Square roots indicate that we should be using"},{"Start":"00:30.690 ","End":"00:35.730","Text":"conjugates and I\u0027m going to remind you what a conjugate is."},{"Start":"00:35.730 ","End":"00:39.345","Text":"If we have an expression A minus B,"},{"Start":"00:39.345 ","End":"00:45.795","Text":"its conjugate is A plus B and vice versa."},{"Start":"00:45.795 ","End":"00:50.975","Text":"We remember the formula that if we multiply these 2 conjugates,"},{"Start":"00:50.975 ","End":"00:53.475","Text":"we get difference of squares,"},{"Start":"00:53.475 ","End":"00:56.910","Text":"A squared minus B squared and this helps because if A"},{"Start":"00:56.910 ","End":"01:01.070","Text":"and/or B is square roots then squaring it gets rid of that."},{"Start":"01:01.070 ","End":"01:04.000","Text":"We\u0027re also going to need as usual,"},{"Start":"01:04.000 ","End":"01:05.690","Text":"and we use this a lot,"},{"Start":"01:05.690 ","End":"01:12.480","Text":"that a number a over infinity could be plus or minus is 0."},{"Start":"01:12.880 ","End":"01:17.030","Text":"Let\u0027s write this as the limit,"},{"Start":"01:17.030 ","End":"01:19.625","Text":"x goes to infinity."},{"Start":"01:19.625 ","End":"01:21.890","Text":"Write it as a fraction,"},{"Start":"01:21.890 ","End":"01:26.120","Text":"where here I have the original expression,"},{"Start":"01:26.120 ","End":"01:32.119","Text":"square root of x squared plus ax minus"},{"Start":"01:32.119 ","End":"01:38.295","Text":"the square root of x squared plus bx,"},{"Start":"01:38.295 ","End":"01:48.180","Text":"and then multiply it by its conjugate square root of x squared plus ax."},{"Start":"01:48.180 ","End":"01:50.715","Text":"Instead of minus, we need a plus,"},{"Start":"01:50.715 ","End":"01:53.850","Text":"plus the square root of x"},{"Start":"01:53.850 ","End":"02:01.939","Text":"squared plus bx and because we\u0027ve multiplied by the conjugate,"},{"Start":"02:01.939 ","End":"02:03.965","Text":"we also have to divide by it,"},{"Start":"02:03.965 ","End":"02:08.330","Text":"to leave our answer unchanged multiply by something over itself,"},{"Start":"02:08.330 ","End":"02:10.775","Text":"it\u0027s 1 that\u0027s okay to do."},{"Start":"02:10.775 ","End":"02:14.555","Text":"So we have here the same expression as here,"},{"Start":"02:14.555 ","End":"02:19.480","Text":"the square root of x squared plus ax,"},{"Start":"02:19.480 ","End":"02:23.060","Text":"the square root of x squared, sorry,"},{"Start":"02:23.060 ","End":"02:25.025","Text":"not minus plus same as here,"},{"Start":"02:25.025 ","End":"02:28.190","Text":"x squared plus bx."},{"Start":"02:28.190 ","End":"02:31.730","Text":"This is going to equal limit,"},{"Start":"02:31.730 ","End":"02:33.920","Text":"and using this formula,"},{"Start":"02:33.920 ","End":"02:35.660","Text":"A squared minus B squared,"},{"Start":"02:35.660 ","End":"02:41.870","Text":"we get x squared plus ax minus that\u0027s the b squared part."},{"Start":"02:41.870 ","End":"02:45.705","Text":"That\u0027s x squared plus bx."},{"Start":"02:45.705 ","End":"02:47.550","Text":"But I need brackets here,"},{"Start":"02:47.550 ","End":"02:53.119","Text":"all over square root of x squared plus ax"},{"Start":"02:53.119 ","End":"03:01.325","Text":"plus the square root of x squared plus b times x."},{"Start":"03:01.325 ","End":"03:04.490","Text":"Now, the numerator,"},{"Start":"03:04.490 ","End":"03:06.800","Text":"the x squared cancels,"},{"Start":"03:06.800 ","End":"03:12.005","Text":"and all we\u0027re left with is ax minus bx."},{"Start":"03:12.005 ","End":"03:17.120","Text":"If we do the algebra and in the denominator,"},{"Start":"03:17.120 ","End":"03:20.570","Text":"we\u0027ll take x squared outside the brackets,"},{"Start":"03:20.570 ","End":"03:30.570","Text":"so we get the square root of x squared times 1 plus a over x,"},{"Start":"03:30.570 ","End":"03:33.780","Text":"plus the square root,"},{"Start":"03:33.780 ","End":"03:41.495","Text":"and here we also take x squared outside the bracket times 1 plus b over x,"},{"Start":"03:41.495 ","End":"03:44.645","Text":"and x goes to infinity."},{"Start":"03:44.645 ","End":"03:50.510","Text":"When I point out that the square root of x squared is the absolute value of x."},{"Start":"03:50.510 ","End":"03:52.400","Text":"It could be plus or minus x."},{"Start":"03:52.400 ","End":"03:56.510","Text":"But because x is going to infinity, it\u0027s positive,"},{"Start":"03:56.510 ","End":"04:02.780","Text":"so the answer is just x and not minus x. I also want to remind you that"},{"Start":"04:02.780 ","End":"04:06.740","Text":"the square root of ab is"},{"Start":"04:06.740 ","End":"04:11.975","Text":"the square root of a times the square root of b if a and b are both positive,"},{"Start":"04:11.975 ","End":"04:15.215","Text":"and so what we\u0027re left with here is the limit,"},{"Start":"04:15.215 ","End":"04:18.450","Text":"x goes to infinity of."},{"Start":"04:18.450 ","End":"04:26.915","Text":"Now we can take x outside here and be left with a minus b, and here,"},{"Start":"04:26.915 ","End":"04:29.840","Text":"if we use this formula here to break it up,"},{"Start":"04:29.840 ","End":"04:32.690","Text":"we get square root of x"},{"Start":"04:32.690 ","End":"04:40.805","Text":"squared times square root of 1 plus a over x,"},{"Start":"04:40.805 ","End":"04:47.615","Text":"plus the square root of 1 plus b over x."},{"Start":"04:47.615 ","End":"04:50.225","Text":"Now I can close the brackets."},{"Start":"04:50.225 ","End":"04:53.780","Text":"Like we said, square root of x squared is x,"},{"Start":"04:53.780 ","End":"04:56.230","Text":"so this cancels with this,"},{"Start":"04:56.230 ","End":"05:00.110","Text":"and now we can substitute x equals infinity,"},{"Start":"05:00.110 ","End":"05:05.909","Text":"so we get a minus b"},{"Start":"05:05.909 ","End":"05:12.085","Text":"over the square root of 1 plus 0,"},{"Start":"05:12.085 ","End":"05:16.820","Text":"plus again the square root of 1 plus 0."},{"Start":"05:16.820 ","End":"05:20.810","Text":"Now the denominator, square root of 1 is 1,"},{"Start":"05:20.810 ","End":"05:22.520","Text":"1 plus 1 is 2."},{"Start":"05:22.520 ","End":"05:27.785","Text":"We\u0027re left with a minus b over 2,"},{"Start":"05:27.785 ","End":"05:31.440","Text":"and this is our answer."}],"ID":4778}],"Thumbnail":null,"ID":3700},{"Name":"Technique 6 Eulers Limit","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Euler\u0027s Limit","Duration":"6m 42s","ChapterTopicVideoID":4771,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/4771.jpeg","UploadDate":"2019-11-14T06:49:59.0000000","DurationForVideoObject":"PT6M42S","Description":null,"MetaTitle":"Euler\u0027s Limit: Video + Workbook | Proprep","MetaDescription":"The Limit of a Function - Technique 6 Eulers Limit. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/the-limit-of-a-function/technique-6-eulers-limit/vid4779","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this clip, we talk about Euler\u0027s limit,"},{"Start":"00:03.630 ","End":"00:06.690","Text":"which is also Technique Number 6 for solving limit,"},{"Start":"00:06.690 ","End":"00:08.940","Text":"and what it states is"},{"Start":"00:08.940 ","End":"00:16.770","Text":"this limit is equal to e where e is that famous number 2.718, etc."},{"Start":"00:16.770 ","End":"00:19.260","Text":"Now, I\u0027d like to generalize this a bit because this is"},{"Start":"00:19.260 ","End":"00:22.140","Text":"just one single limit and we want to solve many limits."},{"Start":"00:22.140 ","End":"00:26.855","Text":"I want to look upon this as a template where the x could be something else."},{"Start":"00:26.855 ","End":"00:29.270","Text":"What I mean is I\u0027d like to write that the"},{"Start":"00:29.270 ","End":"00:37.430","Text":"limit as x goes to infinity of 1 plus 1 over box to the power of box"},{"Start":"00:37.430 ","End":"00:41.990","Text":"is also equal to e provided that this thing goes to infinity."},{"Start":"00:41.990 ","End":"00:44.645","Text":"It\u0027s some function of x which also goes to infinity."},{"Start":"00:44.645 ","End":"00:48.770","Text":"For example, the limit as x goes to infinity."},{"Start":"00:48.770 ","End":"00:51.350","Text":"If instead of x or instead of box,"},{"Start":"00:51.350 ","End":"00:56.910","Text":"I put 2x and make it to the power of 2x that will also equal e."},{"Start":"00:56.910 ","End":"00:59.450","Text":"The important thing is that this and this are"},{"Start":"00:59.450 ","End":"01:02.770","Text":"the same quantity and that this goes to infinity."},{"Start":"01:02.770 ","End":"01:04.100","Text":"Certainly, if x goes to infinity,"},{"Start":"01:04.100 ","End":"01:05.480","Text":"2x goes to infinity."},{"Start":"01:05.480 ","End":"01:07.910","Text":"So another example would be that the"},{"Start":"01:07.910 ","End":"01:12.710","Text":"limit as x goes to infinity of 1 plus 1 over,"},{"Start":"01:12.710 ","End":"01:16.850","Text":"let\u0027s say, x squared plus 1 to the power of x squared plus 1."},{"Start":"01:16.850 ","End":"01:18.070","Text":"See? The same thing,"},{"Start":"01:18.070 ","End":"01:19.760","Text":"and it goes to infinity."},{"Start":"01:19.760 ","End":"01:22.490","Text":"When x goes to infinity, so this is also e."},{"Start":"01:22.490 ","End":"01:30.815","Text":"Limit as x goes to infinity of 1 plus 1 over 4x plus 10"},{"Start":"01:30.815 ","End":"01:34.345","Text":"to the power of 4x plus 10."},{"Start":"01:34.345 ","End":"01:36.935","Text":"This is the same, goes to infinity,"},{"Start":"01:36.935 ","End":"01:38.310","Text":"so this is also e,"},{"Start":"01:38.310 ","End":"01:40.700","Text":"and I think you get the idea by now."},{"Start":"01:40.700 ","End":"01:44.330","Text":"We generalized from Euler\u0027s limit to"},{"Start":"01:44.330 ","End":"01:49.865","Text":"a template from which we got this example and this example and this example,"},{"Start":"01:49.865 ","End":"01:53.105","Text":"but on the exam or even in the homework question,"},{"Start":"01:53.105 ","End":"01:55.550","Text":"you\u0027re not going to be given something like this;"},{"Start":"01:55.550 ","End":"01:58.700","Text":"1 plus 1 over 2x to the 2x because it\u0027s too easy."},{"Start":"01:58.700 ","End":"02:00.110","Text":"You\u0027ll straight away say e."},{"Start":"02:00.110 ","End":"02:02.600","Text":"What we\u0027re going to do is generalize it still further"},{"Start":"02:02.600 ","End":"02:04.070","Text":"and not have something like this,"},{"Start":"02:04.070 ","End":"02:05.390","Text":"but something more general."},{"Start":"02:05.390 ","End":"02:07.295","Text":"I\u0027ll show you some examples in a moment."},{"Start":"02:07.295 ","End":"02:10.280","Text":"The common thing about them all, or if you\u0027d like,"},{"Start":"02:10.280 ","End":"02:15.290","Text":"the indication of when to try to use Euler\u0027s limit is when you"},{"Start":"02:15.290 ","End":"02:18.770","Text":"identify a limit of the form 1 to"},{"Start":"02:18.770 ","End":"02:22.655","Text":"the power of infinity which is one of those undefined forms."},{"Start":"02:22.655 ","End":"02:24.035","Text":"I\u0027ll give you an example."},{"Start":"02:24.035 ","End":"02:33.495","Text":"Say, we have the limit as x goes to infinity of 1 plus 1 over 2x to the power of x."},{"Start":"02:33.495 ","End":"02:38.780","Text":"It doesn\u0027t fit the template of 1 plus 1 over box to the power of box,"},{"Start":"02:38.780 ","End":"02:41.900","Text":"it is of the form 1 to the power of infinity,"},{"Start":"02:41.900 ","End":"02:43.670","Text":"because as x goes to infinity,"},{"Start":"02:43.670 ","End":"02:44.690","Text":"this goes to 0,"},{"Start":"02:44.690 ","End":"02:46.400","Text":"1 plus 0 is 1,"},{"Start":"02:46.400 ","End":"02:48.725","Text":"so here we have 1 and x goes to infinity."},{"Start":"02:48.725 ","End":"02:51.170","Text":"What it means is that you should try"},{"Start":"02:51.170 ","End":"02:54.599","Text":"Euler\u0027s limit after you\u0027ve done some algebraic manipulations,"},{"Start":"02:54.599 ","End":"02:56.940","Text":"somehow force fit it to this template,"},{"Start":"02:56.940 ","End":"03:00.330","Text":"and we\u0027ll see when we solve the example just what that means."},{"Start":"03:00.330 ","End":"03:02.600","Text":"There\u0027s no guarantee that if you have a 1 to"},{"Start":"03:02.600 ","End":"03:05.510","Text":"the power of infinity, that Euler\u0027s limit will help,"},{"Start":"03:05.510 ","End":"03:07.895","Text":"but there\u0027s a good chance and it\u0027s worth trying."},{"Start":"03:07.895 ","End":"03:12.410","Text":"Let me give you some other examples of when it smells like Euler\u0027s limit,"},{"Start":"03:12.410 ","End":"03:13.909","Text":"I mean, 1 to the infinity."},{"Start":"03:13.909 ","End":"03:18.140","Text":"Let\u0027s say, we have limit as x goes to infinity,"},{"Start":"03:18.140 ","End":"03:24.855","Text":"1 plus 1 over 2x to the power of x squared plus 1."},{"Start":"03:24.855 ","End":"03:27.980","Text":"Again, this does not equal to this,"},{"Start":"03:27.980 ","End":"03:31.775","Text":"but it still fits the form 1 to the power of infinity,"},{"Start":"03:31.775 ","End":"03:37.355","Text":"because this goes to 1 as before and this thing goes to infinity as x goes to infinity."},{"Start":"03:37.355 ","End":"03:41.915","Text":"Once again, we know that we might be able to use Euler\u0027s limit here."},{"Start":"03:41.915 ","End":"03:45.380","Text":"A last example of when it looks like we should"},{"Start":"03:45.380 ","End":"03:49.580","Text":"use Euler\u0027s limit or there\u0027s every indication that it might help,"},{"Start":"03:49.580 ","End":"03:51.890","Text":"would be something completely different."},{"Start":"03:51.890 ","End":"03:55.385","Text":"Up till now, it\u0027s looked like it\u0027s 1 plus 1 over something,"},{"Start":"03:55.385 ","End":"03:58.140","Text":"but this time, I\u0027m going to give you something that doesn\u0027t even look like it;"},{"Start":"03:58.140 ","End":"04:04.890","Text":"x squared plus x plus 1 over x squared plus 2x plus 4,"},{"Start":"04:04.890 ","End":"04:08.160","Text":"all this to the power of 2x plus 1."},{"Start":"04:08.160 ","End":"04:11.210","Text":"This completely does not look like the template."},{"Start":"04:11.210 ","End":"04:13.985","Text":"Nevertheless, if you figure out the limit,"},{"Start":"04:13.985 ","End":"04:16.580","Text":"we\u0027ve done this exercise before."},{"Start":"04:16.580 ","End":"04:18.500","Text":"Polynomial over a polynomial,"},{"Start":"04:18.500 ","End":"04:20.120","Text":"you just look at the highest powers;"},{"Start":"04:20.120 ","End":"04:21.290","Text":"x squared over x squared,"},{"Start":"04:21.290 ","End":"04:23.075","Text":"it\u0027s 1, the limit."},{"Start":"04:23.075 ","End":"04:24.830","Text":"Here, as x goes to infinity,"},{"Start":"04:24.830 ","End":"04:26.735","Text":"2x plus 1 also goes to infinity."},{"Start":"04:26.735 ","End":"04:29.405","Text":"Again, we have 1 to the power of infinity,"},{"Start":"04:29.405 ","End":"04:32.690","Text":"which means that we might be able to use"},{"Start":"04:32.690 ","End":"04:37.370","Text":"Euler\u0027s limit after we\u0027ve done some algebraic manipulation and trickery and so forth."},{"Start":"04:37.370 ","End":"04:40.025","Text":"The idea is to bring the limit to a limit which is"},{"Start":"04:40.025 ","End":"04:45.260","Text":"essentially limit as x goes to infinity of our template above."},{"Start":"04:45.260 ","End":"04:47.345","Text":"Essentially, there could be extra bits."},{"Start":"04:47.345 ","End":"04:50.780","Text":"It might be 2 times this or this thing to the power of something,"},{"Start":"04:50.780 ","End":"04:54.200","Text":"but at its essence, it will contain a limit of this form,"},{"Start":"04:54.200 ","End":"04:55.910","Text":"and then we\u0027ll replace this with e."},{"Start":"04:55.910 ","End":"04:58.110","Text":"It\u0027ll become clearer when I do the example."},{"Start":"04:58.110 ","End":"05:00.000","Text":"Let\u0027s take a very simple one."},{"Start":"05:00.000 ","End":"05:07.815","Text":"Limit as x goes to infinity of 1 plus 1 over 4x to the power of x."},{"Start":"05:07.815 ","End":"05:12.110","Text":"Now one way of going about this is what I call spoiling and fixing."},{"Start":"05:12.110 ","End":"05:20.470","Text":"I\u0027ll write this as the limit as x goes to infinity of 1 plus 1 over 4x to the power of,"},{"Start":"05:20.470 ","End":"05:21.800","Text":"now what would I like here?"},{"Start":"05:21.800 ","End":"05:22.850","Text":"These two are not the same."},{"Start":"05:22.850 ","End":"05:24.995","Text":"I would like it to be 4x,"},{"Start":"05:24.995 ","End":"05:29.630","Text":"because then I have the same thing here as here,"},{"Start":"05:29.630 ","End":"05:34.010","Text":"but I\u0027ve now gone and spoiled it because I can\u0027t just throw in a 4 here,"},{"Start":"05:34.010 ","End":"05:35.320","Text":"it\u0027s not the same exercise,"},{"Start":"05:35.320 ","End":"05:37.350","Text":"so now I have to fix what I\u0027ve spoiled,"},{"Start":"05:37.350 ","End":"05:42.330","Text":"and what I\u0027ll do is I\u0027ll write this to the power of 1 over 4,"},{"Start":"05:42.330 ","End":"05:45.005","Text":"because from the rules of algebra,"},{"Start":"05:45.005 ","End":"05:51.125","Text":"a to the power of b to the power of c is a to the power of bc."},{"Start":"05:51.125 ","End":"05:53.525","Text":"This rules of exponents,"},{"Start":"05:53.525 ","End":"05:55.325","Text":"so I can see that this thing,"},{"Start":"05:55.325 ","End":"05:58.310","Text":"if I take 4x times 1/4, will give me the x,"},{"Start":"05:58.310 ","End":"05:59.750","Text":"so I haven\u0027t changed anything."},{"Start":"05:59.750 ","End":"06:07.520","Text":"But now this thing inside the brackets is 1 plus 1 over box to the power of box,"},{"Start":"06:07.520 ","End":"06:11.810","Text":"now, I can replace this as e,"},{"Start":"06:11.810 ","End":"06:13.845","Text":"because that\u0027s what it equals."},{"Start":"06:13.845 ","End":"06:17.055","Text":"This is now e to the power of 1/4,"},{"Start":"06:17.055 ","End":"06:21.135","Text":"and that solves the exercise just by a slight manipulation."},{"Start":"06:21.135 ","End":"06:22.460","Text":"So now you get the idea."},{"Start":"06:22.460 ","End":"06:26.500","Text":"I took a relatively easy example and went over it pretty quickly,"},{"Start":"06:26.500 ","End":"06:30.035","Text":"but there are exercises following this theoretical part."},{"Start":"06:30.035 ","End":"06:32.730","Text":"There it will be explained in much more detail"},{"Start":"06:32.730 ","End":"06:35.840","Text":"and also how to do more difficult ones like this"},{"Start":"06:35.840 ","End":"06:38.825","Text":"which don\u0027t look at all like the template."},{"Start":"06:38.825 ","End":"06:41.090","Text":"Go and do the exercises and enjoy."},{"Start":"06:41.090 ","End":"06:43.110","Text":"We\u0027re done here."}],"ID":4779},{"Watched":false,"Name":"Exercise 1","Duration":"3m 33s","ChapterTopicVideoID":4772,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.885","Text":"In this exercise, we have to compute the limit as"},{"Start":"00:03.885 ","End":"00:08.250","Text":"x goes to minus infinity of this thing here."},{"Start":"00:08.250 ","End":"00:11.700","Text":"If we substitute x equals minus infinity,"},{"Start":"00:11.700 ","End":"00:14.970","Text":"what we get is 1 to the power of minus infinity,"},{"Start":"00:14.970 ","End":"00:18.795","Text":"which is the same as 1 to the power of infinity."},{"Start":"00:18.795 ","End":"00:23.295","Text":"This is another one of those indeterminate undefined forms"},{"Start":"00:23.295 ","End":"00:25.770","Text":"of this to the power of this could be anything."},{"Start":"00:25.770 ","End":"00:27.889","Text":"We have to use another technique."},{"Start":"00:27.889 ","End":"00:31.360","Text":"The most obvious thing to do is to use Euler\u0027s formula."},{"Start":"00:31.360 ","End":"00:34.110","Text":"I like to remind you of what that is."},{"Start":"00:34.110 ","End":"00:36.030","Text":"That is that the limit,"},{"Start":"00:36.030 ","End":"00:37.640","Text":"usually we use the letter x,"},{"Start":"00:37.640 ","End":"00:39.280","Text":"but it could be anything."},{"Start":"00:39.280 ","End":"00:49.175","Text":"I\u0027ll just call it a square as it goes to infinity of 1 plus 1 over square to the power of"},{"Start":"00:49.175 ","End":"00:53.255","Text":"square is equal to the number"},{"Start":"00:53.255 ","End":"01:00.965","Text":"e. The important thing is that this should be the same square everywhere."},{"Start":"01:00.965 ","End":"01:06.020","Text":"This thing has to be the same as this and as this."},{"Start":"01:06.020 ","End":"01:11.220","Text":"Another thing is, I notice that here I have minus infinity."},{"Start":"01:11.220 ","End":"01:18.275","Text":"Lesser known formula is that the same thing works also with minus infinity."},{"Start":"01:18.275 ","End":"01:22.430","Text":"If something goes to minus infinity,"},{"Start":"01:22.430 ","End":"01:31.055","Text":"again,1 plus 1 over something to the power of the same something is also equal to e,"},{"Start":"01:31.055 ","End":"01:33.430","Text":"as long as figures are the same."},{"Start":"01:33.430 ","End":"01:41.960","Text":"In our case, what I suggest is that we use the 2x thing as the square."},{"Start":"01:41.960 ","End":"01:43.940","Text":"If we do that,"},{"Start":"01:43.940 ","End":"01:50.089","Text":"we can say that this thing equals the limit as x goes"},{"Start":"01:50.089 ","End":"01:56.570","Text":"to minus infinity of 1 plus 1 over 2x."},{"Start":"01:56.570 ","End":"02:01.080","Text":"Now, what I\u0027d like to do is to have 2x here."},{"Start":"02:01.080 ","End":"02:03.435","Text":"But this is not what I have."},{"Start":"02:03.435 ","End":"02:07.850","Text":"I can make a correction there and raise this to"},{"Start":"02:07.850 ","End":"02:13.325","Text":"the power of something that will make this just equal to x."},{"Start":"02:13.325 ","End":"02:16.530","Text":"Now, by the laws of exponents,"},{"Start":"02:17.080 ","End":"02:23.880","Text":"a^b^c is a^bc."},{"Start":"02:23.880 ","End":"02:25.370","Text":"That\u0027s from algebra."},{"Start":"02:25.370 ","End":"02:31.850","Text":"If b is 2x and bc has got to be just x,"},{"Start":"02:31.850 ","End":"02:34.100","Text":"then c would be 1/2."},{"Start":"02:34.100 ","End":"02:38.090","Text":"In other words, if I take this to the power of 1/2,"},{"Start":"02:38.090 ","End":"02:40.865","Text":"if I multiply exponents 2x times 1/2,"},{"Start":"02:40.865 ","End":"02:42.590","Text":"that gives us just x."},{"Start":"02:42.590 ","End":"02:44.870","Text":"There is another small point I just would like to"},{"Start":"02:44.870 ","End":"02:49.580","Text":"mention is that we do have three places where square appears here,"},{"Start":"02:49.580 ","End":"02:51.110","Text":"here, but also here."},{"Start":"02:51.110 ","End":"02:54.710","Text":"Now, notice that if x goes to minus infinity,"},{"Start":"02:54.710 ","End":"02:57.680","Text":"then 2x also goes to minus infinity."},{"Start":"02:57.680 ","End":"03:01.150","Text":"So it doesn\u0027t matter the factor of 2."},{"Start":"03:01.150 ","End":"03:06.030","Text":"We\u0027re all set now for completing the exercise."},{"Start":"03:06.030 ","End":"03:09.800","Text":"We have the limit of something to the power of 1/2."},{"Start":"03:09.800 ","End":"03:14.390","Text":"We can take the limit of this thing separately then make it to the power of 1/2."},{"Start":"03:14.390 ","End":"03:19.680","Text":"This thing is actually using the second formula with square being"},{"Start":"03:19.680 ","End":"03:26.825","Text":"2x gives us e. This is just equal to e to the power of 1/2,"},{"Start":"03:26.825 ","End":"03:30.760","Text":"which is the square root of e,"},{"Start":"03:30.760 ","End":"03:34.000","Text":"and that\u0027s the answer."}],"ID":4780},{"Watched":false,"Name":"Exercise 2","Duration":"3m 10s","ChapterTopicVideoID":4774,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this exercise, we have to find the limit as x goes"},{"Start":"00:03.600 ","End":"00:07.200","Text":"to minus infinity of this expression."},{"Start":"00:07.200 ","End":"00:10.590","Text":"A straightforward substitution is what we would try first."},{"Start":"00:10.590 ","End":"00:13.350","Text":"But if we put x as minus infinity,"},{"Start":"00:13.350 ","End":"00:18.359","Text":"you quickly see that what we get is 1 to the power of infinity,"},{"Start":"00:18.359 ","End":"00:21.510","Text":"which is another one of those undefined expressions."},{"Start":"00:21.510 ","End":"00:24.360","Text":"1 to the power of infinity does not make sense."},{"Start":"00:24.360 ","End":"00:29.145","Text":"So we have to use another technique and the obvious thing is to use Euler\u0027s formula."},{"Start":"00:29.145 ","End":"00:31.440","Text":"Looks very much like it."},{"Start":"00:31.440 ","End":"00:36.070","Text":"Let me remind you what Euler\u0027s formula is."},{"Start":"00:36.070 ","End":"00:39.060","Text":"Limit, it\u0027s usually written with x,"},{"Start":"00:39.060 ","End":"00:44.920","Text":"but it could be any letter or any quantity goes to infinity"},{"Start":"00:44.920 ","End":"00:52.360","Text":"of 1 plus 1 over the same thing to the power of that same thing."},{"Start":"00:52.360 ","End":"00:55.255","Text":"This is equal to e,"},{"Start":"00:55.255 ","End":"00:58.045","Text":"and that\u0027s Euler\u0027s formula."},{"Start":"00:58.045 ","End":"01:03.414","Text":"In our case, what I want to do is try and use Euler\u0027s formula"},{"Start":"01:03.414 ","End":"01:09.655","Text":"with this undefined square as being x to the power of 2."},{"Start":"01:09.655 ","End":"01:13.360","Text":"What we\u0027re going to do is say that\u0027s equals,"},{"Start":"01:13.360 ","End":"01:15.505","Text":"and I want to somehow use this."},{"Start":"01:15.505 ","End":"01:20.485","Text":"It\u0027s the limit as x goes to"},{"Start":"01:20.485 ","End":"01:26.675","Text":"minus infinity of 1 plus 1 over x squared."},{"Start":"01:26.675 ","End":"01:32.525","Text":"Now what I need here is x squared to make it look like this."},{"Start":"01:32.525 ","End":"01:35.810","Text":"So I\u0027m going to have to fix this because I don\u0027t have x squared,"},{"Start":"01:35.810 ","End":"01:41.290","Text":"l only have x. I\u0027m going to fix it by putting here a 1 over x."},{"Start":"01:41.290 ","End":"01:42.950","Text":"Where did that come from?"},{"Start":"01:42.950 ","End":"01:46.670","Text":"Well, there\u0027s the algebraic rule that a to the power of"},{"Start":"01:46.670 ","End":"01:51.845","Text":"b to the power of c is a to the power of bc,"},{"Start":"01:51.845 ","End":"01:53.960","Text":"and b here is x squared."},{"Start":"01:53.960 ","End":"01:58.430","Text":"I asked myself, x squared times what will give me x?"},{"Start":"01:58.430 ","End":"02:01.340","Text":"The answer comes out to be 1 over x."},{"Start":"02:01.340 ","End":"02:02.900","Text":"Now here\u0027s the other thing."},{"Start":"02:02.900 ","End":"02:05.150","Text":"This thing appears in 3 places,"},{"Start":"02:05.150 ","End":"02:06.830","Text":"here, here, and here."},{"Start":"02:06.830 ","End":"02:09.605","Text":"When x goes to minus infinity,"},{"Start":"02:09.605 ","End":"02:14.750","Text":"actually x squared goes to plus infinity."},{"Start":"02:14.750 ","End":"02:17.690","Text":"I just write the plus for emphasis."},{"Start":"02:17.690 ","End":"02:20.210","Text":"I have 2 limits,"},{"Start":"02:20.210 ","End":"02:21.230","Text":"I have 2 parts."},{"Start":"02:21.230 ","End":"02:23.135","Text":"I have this part here."},{"Start":"02:23.135 ","End":"02:29.210","Text":"In this part, I\u0027m going to let x squared go to infinity."},{"Start":"02:29.210 ","End":"02:32.495","Text":"I also have this part here and here,"},{"Start":"02:32.495 ","End":"02:35.480","Text":"x goes to minus infinity,"},{"Start":"02:35.480 ","End":"02:37.385","Text":"which was what the original was."},{"Start":"02:37.385 ","End":"02:41.345","Text":"So what we\u0027re going to get is that this equals,"},{"Start":"02:41.345 ","End":"02:44.615","Text":"now this part here is exactly this formula"},{"Start":"02:44.615 ","End":"02:48.380","Text":"with x squared being what I highlighted in yellow."},{"Start":"02:48.380 ","End":"02:53.810","Text":"This part is going to go to e. This part here,"},{"Start":"02:53.810 ","End":"02:59.275","Text":"we have 1 over minus infinity and so that\u0027s equal to 0."},{"Start":"02:59.275 ","End":"03:01.235","Text":"0 minus if you like,"},{"Start":"03:01.235 ","End":"03:05.930","Text":"but 0 minus is just the same as 0."},{"Start":"03:05.930 ","End":"03:08.140","Text":"So this is equal to 1."},{"Start":"03:08.140 ","End":"03:10.780","Text":"That\u0027s the answer."}],"ID":4782},{"Watched":false,"Name":"Exercise 3","Duration":"3m 34s","ChapterTopicVideoID":4775,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:04.380 ","End":"00:09.690","Text":"infinity of x plus 2 over x to the power of x."},{"Start":"00:09.690 ","End":"00:12.450","Text":"Let me just write this at the side."},{"Start":"00:12.450 ","End":"00:16.230","Text":"Just do a bit of algebra without the limit for the moment."},{"Start":"00:16.230 ","End":"00:18.600","Text":"If I was to substitute x equals infinity,"},{"Start":"00:18.600 ","End":"00:20.880","Text":"which is what we usually do in cases like this,"},{"Start":"00:20.880 ","End":"00:24.630","Text":"the first thing I would get would be infinity"},{"Start":"00:24.630 ","End":"00:28.845","Text":"here over infinity and that would stop us already."},{"Start":"00:28.845 ","End":"00:31.170","Text":"But even if we really can continue,"},{"Start":"00:31.170 ","End":"00:32.845","Text":"but if we do some algebra,"},{"Start":"00:32.845 ","End":"00:37.055","Text":"x plus 2 over x is equal to,"},{"Start":"00:37.055 ","End":"00:38.210","Text":"if I do the division,"},{"Start":"00:38.210 ","End":"00:40.070","Text":"each 1 of these is divided by x,"},{"Start":"00:40.070 ","End":"00:44.135","Text":"so I get 1 plus 2 over x to the power of x."},{"Start":"00:44.135 ","End":"00:47.885","Text":"Then if I substitute x goes to infinity,"},{"Start":"00:47.885 ","End":"00:50.510","Text":"I\u0027ll get 1 plus 2 over infinity,"},{"Start":"00:50.510 ","End":"00:54.440","Text":"which will be 1 and x goes to infinity."},{"Start":"00:54.440 ","End":"00:56.780","Text":"We\u0027re still not much better off."},{"Start":"00:56.780 ","End":"00:59.120","Text":"Infinity over infinity was indeterminate,"},{"Start":"00:59.120 ","End":"01:02.120","Text":"but 1 to the infinity is also undefined."},{"Start":"01:02.120 ","End":"01:05.450","Text":"However, it does indicate Euler\u0027s formula,"},{"Start":"01:05.450 ","End":"01:08.090","Text":"especially when I see something over x to the power of"},{"Start":"01:08.090 ","End":"01:11.855","Text":"x. I\u0027d like to remind you what Euler\u0027s formula is."},{"Start":"01:11.855 ","End":"01:13.280","Text":"He wrote many things,"},{"Start":"01:13.280 ","End":"01:17.895","Text":"but 1 of the things he discovered was that the limit,"},{"Start":"01:17.895 ","End":"01:20.040","Text":"and instead of writing x,"},{"Start":"01:20.040 ","End":"01:21.120","Text":"let\u0027s be more general,"},{"Start":"01:21.120 ","End":"01:23.505","Text":"let\u0027s just write a square meaning anything,"},{"Start":"01:23.505 ","End":"01:32.035","Text":"goes to infinity of 1 plus 1 over the same quantity to the power of the same thing."},{"Start":"01:32.035 ","End":"01:39.690","Text":"This limit is equal to e. Let\u0027s try and use this in our case."},{"Start":"01:39.690 ","End":"01:41.900","Text":"If I rewrite this,"},{"Start":"01:41.900 ","End":"01:47.600","Text":"I\u0027ll get the limit as x goes to"},{"Start":"01:47.600 ","End":"01:55.980","Text":"infinity of 1 plus 2 over x to the power of x."},{"Start":"01:55.980 ","End":"01:59.465","Text":"This is already looking a bit more like this,"},{"Start":"01:59.465 ","End":"02:01.730","Text":"except here we have 1 over something,"},{"Start":"02:01.730 ","End":"02:03.440","Text":"in here we have 2 over some thing,"},{"Start":"02:03.440 ","End":"02:05.930","Text":"so let\u0027s take care of that."},{"Start":"02:05.930 ","End":"02:12.480","Text":"That\u0027s the limit as x goes to infinity of 1 plus,"},{"Start":"02:12.480 ","End":"02:15.150","Text":"now I want 1 over something."},{"Start":"02:15.150 ","End":"02:16.830","Text":"To get this 1,"},{"Start":"02:16.830 ","End":"02:19.135","Text":"I divided the 2 by 2."},{"Start":"02:19.135 ","End":"02:21.260","Text":"If I do something in the numerator,"},{"Start":"02:21.260 ","End":"02:24.635","Text":"I have to divide it likewise in the denominator,"},{"Start":"02:24.635 ","End":"02:26.420","Text":"and that takes care of that."},{"Start":"02:26.420 ","End":"02:29.525","Text":"But I still have a problem is that here there\u0027s x over 2,"},{"Start":"02:29.525 ","End":"02:32.030","Text":"and I\u0027d like an x over 2 here also,"},{"Start":"02:32.030 ","End":"02:33.940","Text":"this is the same as this."},{"Start":"02:33.940 ","End":"02:37.700","Text":"Let\u0027s write it as x over 2."},{"Start":"02:37.700 ","End":"02:41.255","Text":"But now I\u0027ve changed the exercise because the 2 shouldn\u0027t be here."},{"Start":"02:41.255 ","End":"02:46.670","Text":"I\u0027ll compensate and x over 2 times something has to be x,"},{"Start":"02:46.670 ","End":"02:48.395","Text":"so that will be 2."},{"Start":"02:48.395 ","End":"02:55.715","Text":"Now, this is exactly in this form where the square is represented by x over 2."},{"Start":"02:55.715 ","End":"02:59.090","Text":"Notice that when x goes to infinity,"},{"Start":"02:59.090 ","End":"03:01.685","Text":"x over 2, which is what this is,"},{"Start":"03:01.685 ","End":"03:03.580","Text":"also goes to infinity."},{"Start":"03:03.580 ","End":"03:07.880","Text":"Basically, what I\u0027m saying is that I\u0027m going to substitute,"},{"Start":"03:07.880 ","End":"03:13.000","Text":"my square in this case will be represented by x over 2."},{"Start":"03:13.000 ","End":"03:15.245","Text":"I have the square here and here,"},{"Start":"03:15.245 ","End":"03:18.650","Text":"and over here I have it because if x goes to infinity,"},{"Start":"03:18.650 ","End":"03:21.020","Text":"then so does x over 2."},{"Start":"03:21.020 ","End":"03:24.185","Text":"This part here is Euler\u0027s formula."},{"Start":"03:24.185 ","End":"03:29.225","Text":"When I substitute x equals infinity or goes to infinity and I still have the 2 out here."},{"Start":"03:29.225 ","End":"03:34.680","Text":"This is just equal to e to the power of 2 and that\u0027s it."}],"ID":4783},{"Watched":false,"Name":"Exercise 4","Duration":"4m 47s","ChapterTopicVideoID":4776,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.285","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression."},{"Start":"00:06.285 ","End":"00:09.810","Text":"If we try to substitute x equals infinity,"},{"Start":"00:09.810 ","End":"00:12.150","Text":"we get 1 minus 0,"},{"Start":"00:12.150 ","End":"00:16.855","Text":"which is 1^infinity squared minus 1, which is infinity."},{"Start":"00:16.855 ","End":"00:21.335","Text":"In short, we get something of the form 1^infinity,"},{"Start":"00:21.335 ","End":"00:27.050","Text":"which is often indicative of the Euler formula and it looks like the Euler formula."},{"Start":"00:27.050 ","End":"00:31.415","Text":"Let me just remind you what that formula says."},{"Start":"00:31.415 ","End":"00:39.060","Text":"Euler discovered that the limit x goes to infinity of 1 plus 1"},{"Start":"00:39.060 ","End":"00:47.150","Text":"over box to the power of a box is equal to e. In fact,"},{"Start":"00:47.150 ","End":"00:50.780","Text":"it also works for minus infinity."},{"Start":"00:50.780 ","End":"00:55.220","Text":"In other words, this could also be a minus infinity instead of"},{"Start":"00:55.220 ","End":"01:00.920","Text":"a plus infinity and the same formula holds. We might need that."},{"Start":"01:00.920 ","End":"01:07.580","Text":"We\u0027re going to try and get this in this form and for the couple of things to notice,"},{"Start":"01:07.580 ","End":"01:11.270","Text":"1 is that here we have a plus and here we have a minus."},{"Start":"01:11.270 ","End":"01:14.570","Text":"The other is that this and this are not the same,"},{"Start":"01:14.570 ","End":"01:19.040","Text":"whereas this box or this box are meant to be the same quantity."},{"Start":"01:19.040 ","End":"01:22.210","Text":"We\u0027ll somehow get it into shape."},{"Start":"01:22.210 ","End":"01:31.785","Text":"We\u0027ll do is just rewrite it as the limit x goes to infinity and now I\u0027ll write this as 1."},{"Start":"01:31.785 ","End":"01:35.990","Text":"Now to get rid of the minus and make it a plus,"},{"Start":"01:35.990 ","End":"01:39.735","Text":"I\u0027ll write plus but 1 over,"},{"Start":"01:39.735 ","End":"01:42.440","Text":"and I\u0027ll just take the minus into the denominator,"},{"Start":"01:42.440 ","End":"01:46.310","Text":"so minus x squared and that takes care of this part"},{"Start":"01:46.310 ","End":"01:50.405","Text":"and now we have to get this as the same as the denominator."},{"Start":"01:50.405 ","End":"01:54.470","Text":"I\u0027ll start off by writing minus x squared."},{"Start":"01:54.470 ","End":"02:00.280","Text":"Now, I\u0027ve changed the exercise because it should have been x squared minus 1."},{"Start":"02:00.280 ","End":"02:02.355","Text":"I\u0027ll need to compensate."},{"Start":"02:02.355 ","End":"02:11.059","Text":"I\u0027ll take this expression here and ask myself x squared times what will give me this?"},{"Start":"02:11.059 ","End":"02:14.765","Text":"The answer would be this over this."},{"Start":"02:14.765 ","End":"02:19.820","Text":"Just like if I say how many times does 3 goes into 12,"},{"Start":"02:19.820 ","End":"02:22.460","Text":"the answer would be 12 over 3."},{"Start":"02:22.460 ","End":"02:27.110","Text":"I write this as x squared minus 1"},{"Start":"02:27.110 ","End":"02:31.895","Text":"over minus x squared and you can see if you multiply it out,"},{"Start":"02:31.895 ","End":"02:34.985","Text":"this cancels with this and we\u0027ve got back to the original."},{"Start":"02:34.985 ","End":"02:42.320","Text":"What I\u0027m trying to do here is to get it to that this box and"},{"Start":"02:42.320 ","End":"02:45.680","Text":"this box and this box are all going to"},{"Start":"02:45.680 ","End":"02:49.985","Text":"be like the minus x squared here is going to be what it is,"},{"Start":"02:49.985 ","End":"02:52.565","Text":"the same as this minus x squared."},{"Start":"02:52.565 ","End":"03:01.325","Text":"Here\u0027s something, another fine point and that is that x may go to infinity,"},{"Start":"03:01.325 ","End":"03:03.560","Text":"but when x goes to infinity,"},{"Start":"03:03.560 ","End":"03:08.540","Text":"then minus x squared goes to minus infinity."},{"Start":"03:08.540 ","End":"03:10.880","Text":"So what we have,"},{"Start":"03:10.880 ","End":"03:13.970","Text":"if I try to substitute is that on the 1 hand,"},{"Start":"03:13.970 ","End":"03:19.490","Text":"I have this piece together with the minus x squared and this,"},{"Start":"03:19.490 ","End":"03:22.415","Text":"and x minus x squared goes to minus infinity."},{"Start":"03:22.415 ","End":"03:24.680","Text":"Basically, I have a case of"},{"Start":"03:24.680 ","End":"03:30.175","Text":"this formula with the minus infinity case not the plus infinity."},{"Start":"03:30.175 ","End":"03:34.290","Text":"This part becomes e. From here I get the e."},{"Start":"03:34.290 ","End":"03:38.870","Text":"But the other part is this part where I used the other limit,"},{"Start":"03:38.870 ","End":"03:40.715","Text":"x goes to infinity."},{"Start":"03:40.715 ","End":"03:46.775","Text":"In other words, for here I was using x squared goes to minus infinity."},{"Start":"03:46.775 ","End":"03:50.555","Text":"But here I\u0027m using x goes to infinity."},{"Start":"03:50.555 ","End":"03:51.950","Text":"So this part here,"},{"Start":"03:51.950 ","End":"03:53.420","Text":"if x is infinity,"},{"Start":"03:53.420 ","End":"03:55.265","Text":"divide top and bottom."},{"Start":"03:55.265 ","End":"03:57.880","Text":"I\u0027ll just do that here."},{"Start":"03:57.880 ","End":"04:05.060","Text":"I have x squared minus 1 over minus x squared is equal to,"},{"Start":"04:05.060 ","End":"04:07.445","Text":"I\u0027ll take x squared out of the numerator,"},{"Start":"04:07.445 ","End":"04:15.035","Text":"1 minus 1 over x squared and here over minus x squared,"},{"Start":"04:15.035 ","End":"04:16.980","Text":"the x squared cancels,"},{"Start":"04:16.980 ","End":"04:19.295","Text":"and just have a minus here."},{"Start":"04:19.295 ","End":"04:25.985","Text":"This gives me minus 1 plus 1 over x squared."},{"Start":"04:25.985 ","End":"04:29.190","Text":"When x goes to infinity,"},{"Start":"04:29.920 ","End":"04:32.390","Text":"1 over infinity is 0,"},{"Start":"04:32.390 ","End":"04:35.035","Text":"this is just equal to minus 1."},{"Start":"04:35.035 ","End":"04:38.690","Text":"That was the part which I marked this way,"},{"Start":"04:38.690 ","End":"04:40.655","Text":"and this is the part in the circle."},{"Start":"04:40.655 ","End":"04:47.790","Text":"In short, the answer is e^minus 1 or 1 over e. That\u0027s the answer."}],"ID":4784},{"Watched":false,"Name":"Exercise 5","Duration":"7m 23s","ChapterTopicVideoID":4777,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.185","Text":"In this exercise, we want to find the limit as x goes to infinity of this expression."},{"Start":"00:07.185 ","End":"00:12.660","Text":"First thing we do is try putting x equals infinity and see what we get."},{"Start":"00:12.660 ","End":"00:16.290","Text":"2x plus 3 over 2x minus 3,"},{"Start":"00:16.290 ","End":"00:18.390","Text":"if we just straight away substitute infinity,"},{"Start":"00:18.390 ","End":"00:19.970","Text":"we get infinity over infinity,"},{"Start":"00:19.970 ","End":"00:21.690","Text":"but usually we do some algebra."},{"Start":"00:21.690 ","End":"00:25.980","Text":"We take the x outside the brackets here and here the x cancels,"},{"Start":"00:25.980 ","End":"00:28.950","Text":"and what we\u0027re left with is 2 plus 3 over x,"},{"Start":"00:28.950 ","End":"00:30.735","Text":"2 minus 3 over x."},{"Start":"00:30.735 ","End":"00:32.550","Text":"This part goes to 0,"},{"Start":"00:32.550 ","End":"00:33.780","Text":"2 over 2 is 1."},{"Start":"00:33.780 ","End":"00:37.815","Text":"In short, we get 1 to the power of infinity."},{"Start":"00:37.815 ","End":"00:42.515","Text":"Whenever we get something of the form 1 to the infinity,"},{"Start":"00:42.515 ","End":"00:44.870","Text":"first of all, it\u0027s meaningless,"},{"Start":"00:44.870 ","End":"00:47.540","Text":"it\u0027s not defined or sometimes called indeterminate."},{"Start":"00:47.540 ","End":"00:52.820","Text":"Second is often a hint that we should be using Euler\u0027s limit,"},{"Start":"00:52.820 ","End":"00:55.565","Text":"and I\u0027ve pre-written it here."},{"Start":"00:55.565 ","End":"01:00.125","Text":"What Euler\u0027s limit says that if we take the limit of some expression,"},{"Start":"01:00.125 ","End":"01:02.680","Text":"could be x, could be something else,"},{"Start":"01:02.680 ","End":"01:04.520","Text":"as this thing goes to infinity,"},{"Start":"01:04.520 ","End":"01:07.240","Text":"the limit of 1 plus 1 over,"},{"Start":"01:07.240 ","End":"01:12.830","Text":"to the power of box is equal to e. Now we look here and it doesn\u0027t"},{"Start":"01:12.830 ","End":"01:17.960","Text":"really look very much like this expression except that if x goes to infinity,"},{"Start":"01:17.960 ","End":"01:19.700","Text":"we have an x here."},{"Start":"01:19.700 ","End":"01:27.050","Text":"What we do first is make what\u0027s inside the brackets look like 1 plus 1 over something."},{"Start":"01:27.050 ","End":"01:35.745","Text":"Let\u0027s do that at the side and say that we want 2x plus 3 over 2x minus 3."},{"Start":"01:35.745 ","End":"01:42.080","Text":"We want this to equal 1 plus 1 over that something."},{"Start":"01:42.080 ","End":"01:43.550","Text":"Now we do a bit of algebra."},{"Start":"01:43.550 ","End":"01:45.980","Text":"I\u0027ll put this on this side first of all,"},{"Start":"01:45.980 ","End":"01:47.735","Text":"but at the same time,"},{"Start":"01:47.735 ","End":"01:50.255","Text":"I\u0027ll throw the 1 back into the other side,"},{"Start":"01:50.255 ","End":"01:57.765","Text":"which will be 2x plus 3 over 2x minus 3 minus the 1."},{"Start":"01:57.765 ","End":"02:02.885","Text":"This equals, if I write it under a common denominator,"},{"Start":"02:02.885 ","End":"02:07.625","Text":"the common denominator is 2x minus 3, and the numerator,"},{"Start":"02:07.625 ","End":"02:09.080","Text":"we already have what we have,"},{"Start":"02:09.080 ","End":"02:12.305","Text":"which is 2x plus 3, but the minus 1,"},{"Start":"02:12.305 ","End":"02:15.725","Text":"1 is like 2x minus 3 over 2x minus 3,"},{"Start":"02:15.725 ","End":"02:18.215","Text":"a fraction a over a is just 1,"},{"Start":"02:18.215 ","End":"02:21.190","Text":"and here it\u0027s 2 minus 3 over 2x minus 3."},{"Start":"02:21.190 ","End":"02:23.610","Text":"We have minus,"},{"Start":"02:23.610 ","End":"02:28.715","Text":"into brackets there 2x minus 3 it\u0027s in brackets, over this."},{"Start":"02:28.715 ","End":"02:35.145","Text":"Then what this gives us is that this equals the 2x minus 3 on the top,"},{"Start":"02:35.145 ","End":"02:37.560","Text":"2x and 2x cancel out,"},{"Start":"02:37.560 ","End":"02:41.420","Text":"3 minus, minus 3 is 6."},{"Start":"02:41.420 ","End":"02:46.850","Text":"Last step is that box is just the inverse of this."},{"Start":"02:46.850 ","End":"02:49.070","Text":"If we take the reciprocal of this fraction,"},{"Start":"02:49.070 ","End":"02:50.705","Text":"we should do it here too,"},{"Start":"02:50.705 ","End":"02:56.010","Text":"and that gives us 2x minus 3 over 6."},{"Start":"02:56.010 ","End":"02:59.780","Text":"This expression here, I\u0027ll even put it in a box to"},{"Start":"02:59.780 ","End":"03:03.655","Text":"indicate that this is what we found for the box here."},{"Start":"03:03.655 ","End":"03:07.520","Text":"Now we can write this thing 1 step further,"},{"Start":"03:07.520 ","End":"03:13.550","Text":"that this is equal to the limit as x goes to infinity of"},{"Start":"03:13.550 ","End":"03:18.360","Text":"1 plus 1 over 2x minus"},{"Start":"03:18.360 ","End":"03:23.820","Text":"3 over 6 all this to the power of x,"},{"Start":"03:23.820 ","End":"03:25.520","Text":"that\u0027s what we have."},{"Start":"03:25.520 ","End":"03:29.630","Text":"We have now is that we\u0027ve put it in the form of 1 plus 1 over this."},{"Start":"03:29.630 ","End":"03:33.530","Text":"The next step is to make sure that this is the same as this,"},{"Start":"03:33.530 ","End":"03:39.095","Text":"and 1 way to do it is just to write something wrong and then fix it,"},{"Start":"03:39.095 ","End":"03:41.060","Text":"and I\u0027ll show you what I mean."},{"Start":"03:41.060 ","End":"03:45.830","Text":"You put x goes to infinity and we put in some wishful thinking."},{"Start":"03:45.830 ","End":"03:48.980","Text":"Now, I would like it if they were the same,"},{"Start":"03:48.980 ","End":"03:53.165","Text":"so I start off by saying 2x minus 3 over 6,"},{"Start":"03:53.165 ","End":"03:55.460","Text":"and here, like these 2 are equal,"},{"Start":"03:55.460 ","End":"03:59.365","Text":"I put 2x minus 3 over 6,"},{"Start":"03:59.365 ","End":"04:01.130","Text":"but that\u0027s not allowed, that\u0027s cheating."},{"Start":"04:01.130 ","End":"04:02.750","Text":"I\u0027ve changed the exercise."},{"Start":"04:02.750 ","End":"04:07.775","Text":"What I\u0027m going to do is write something else over here that will make it right."},{"Start":"04:07.775 ","End":"04:12.739","Text":"Just want to remind you of a simple rule in algebra in general,"},{"Start":"04:12.739 ","End":"04:20.660","Text":"that a^b^c is a^bc."},{"Start":"04:20.660 ","End":"04:22.160","Text":"If I put something here,"},{"Start":"04:22.160 ","End":"04:25.310","Text":"and this times this equals x, then we\u0027re all right."},{"Start":"04:25.310 ","End":"04:30.010","Text":"I have to ask what multiplied by"},{"Start":"04:30.010 ","End":"04:35.620","Text":"2x minus 3 over 6 is going to give us x,"},{"Start":"04:35.620 ","End":"04:39.739","Text":"and then it will be fine because of this formula that we multiply."},{"Start":"04:39.739 ","End":"04:42.260","Text":"What we do to find this question mark,"},{"Start":"04:42.260 ","End":"04:44.755","Text":"is bring these 2 over the other side."},{"Start":"04:44.755 ","End":"04:50.300","Text":"Our question mark is just 6 times x over this,"},{"Start":"04:50.300 ","End":"04:55.170","Text":"and that means that if I put that here,"},{"Start":"04:55.170 ","End":"04:59.235","Text":"6x over 2x minus 3,"},{"Start":"04:59.235 ","End":"05:00.390","Text":"that now it\u0027s fine."},{"Start":"05:00.390 ","End":"05:02.530","Text":"Before I had this,"},{"Start":"05:02.530 ","End":"05:04.960","Text":"I\u0027ve just changed the exercise, it was wrong,"},{"Start":"05:04.960 ","End":"05:08.230","Text":"but I put this to make it right again."},{"Start":"05:08.230 ","End":"05:12.790","Text":"Here I have the same thing at the top here."},{"Start":"05:12.790 ","End":"05:22.430","Text":"The only thing that isn\u0027t quite right is that here I have x instead of 2x minus 3 over 6."},{"Start":"05:22.430 ","End":"05:26.620","Text":"But what I say is that it doesn\u0027t matter because if x"},{"Start":"05:26.620 ","End":"05:31.090","Text":"goes to infinity and 2x minus 3 over 6 also goes to infinity."},{"Start":"05:31.090 ","End":"05:35.390","Text":"I mean twice infinity minus 3 is also infinity and vice versa."},{"Start":"05:35.390 ","End":"05:38.045","Text":"I could indicate that,"},{"Start":"05:38.045 ","End":"05:39.950","Text":"yes, x goes to infinity,"},{"Start":"05:39.950 ","End":"05:41.585","Text":"but at the same time,"},{"Start":"05:41.585 ","End":"05:46.820","Text":"we also have that box also goes to infinity."},{"Start":"05:46.820 ","End":"05:48.260","Text":"Now we have this,"},{"Start":"05:48.260 ","End":"05:49.550","Text":"this and this are equal,"},{"Start":"05:49.550 ","End":"05:52.010","Text":"this, this and this are equal."},{"Start":"05:52.010 ","End":"05:56.830","Text":"We can use Euler\u0027s limit on this inside part,"},{"Start":"05:56.830 ","End":"05:59.280","Text":"and what we\u0027re left with is the limit,"},{"Start":"05:59.280 ","End":"06:01.910","Text":"and this time I\u0027ve used up the box in this limit,"},{"Start":"06:01.910 ","End":"06:03.290","Text":"so I just need the x,"},{"Start":"06:03.290 ","End":"06:05.405","Text":"x goes to infinity,"},{"Start":"06:05.405 ","End":"06:07.750","Text":"all this becomes just e,"},{"Start":"06:07.750 ","End":"06:14.375","Text":"and we still have the 6x over 2x minus 3."},{"Start":"06:14.375 ","End":"06:18.260","Text":"Now all we have to do is find the limit as x goes to infinity of this part,"},{"Start":"06:18.260 ","End":"06:21.335","Text":"and the answer will be e to the power of it."},{"Start":"06:21.335 ","End":"06:27.545","Text":"What we have is that 6x over 2x minus 3,"},{"Start":"06:27.545 ","End":"06:31.205","Text":"our usual technique of taking out the highest power of x."},{"Start":"06:31.205 ","End":"06:36.180","Text":"This gives us x times 6, and here,"},{"Start":"06:36.180 ","End":"06:40.395","Text":"x times 2 minus 3 over x over,"},{"Start":"06:40.395 ","End":"06:41.630","Text":"our x is going to infinity,"},{"Start":"06:41.630 ","End":"06:42.830","Text":"so it\u0027s not infinity,"},{"Start":"06:42.830 ","End":"06:45.020","Text":"so we cancel by it."},{"Start":"06:45.020 ","End":"06:48.115","Text":"I\u0027m just working on this fraction part,"},{"Start":"06:48.115 ","End":"06:57.485","Text":"and this is equal to 6 over 2 minus 3 over x."},{"Start":"06:57.485 ","End":"06:59.990","Text":"Now when x goes to infinity,"},{"Start":"06:59.990 ","End":"07:01.640","Text":"3 over infinity is 0,"},{"Start":"07:01.640 ","End":"07:03.685","Text":"so we get 6 over 2,"},{"Start":"07:03.685 ","End":"07:10.160","Text":"and this equals 3 when x goes to infinity."},{"Start":"07:10.160 ","End":"07:14.345","Text":"Now all I have to do is put that back in here and say that this"},{"Start":"07:14.345 ","End":"07:19.065","Text":"equals e to the power of 3 because that\u0027s what this thing is,"},{"Start":"07:19.065 ","End":"07:20.550","Text":"and that\u0027s the answer,"},{"Start":"07:20.550 ","End":"07:23.410","Text":"e cubed, we\u0027re done."}],"ID":4785},{"Watched":false,"Name":"Exercise 6","Duration":"2m 56s","ChapterTopicVideoID":4778,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.120","Text":"In this exercise, we have to find the limit as x goes to 0 of 1 plus sine x^1 over x."},{"Start":"00:06.120 ","End":"00:11.625","Text":"First thing we can do is try substituting x equals 0 and what we get is,"},{"Start":"00:11.625 ","End":"00:16.290","Text":"1 plus 0 is 1^1 over 0,"},{"Start":"00:16.290 ","End":"00:20.460","Text":"which is plus or minus infinity depending on whether we use a left or right limit."},{"Start":"00:20.460 ","End":"00:25.590","Text":"In short, what we get here is 1 to the power of infinity,"},{"Start":"00:25.590 ","End":"00:26.985","Text":"whether it\u0027s plus or minus,"},{"Start":"00:26.985 ","End":"00:30.090","Text":"it\u0027s undefined, it\u0027s an indeterminate expression."},{"Start":"00:30.090 ","End":"00:33.520","Text":"We have to use something from our bag of tricks."},{"Start":"00:33.520 ","End":"00:35.434","Text":"When we get 1 to the infinity,"},{"Start":"00:35.434 ","End":"00:38.615","Text":"it\u0027s very often an indicator of Euler\u0027s limit."},{"Start":"00:38.615 ","End":"00:40.400","Text":"Euler\u0027s limit comes in"},{"Start":"00:40.400 ","End":"00:44.605","Text":"various forms and I\u0027d like to write the form that we\u0027re going to use."},{"Start":"00:44.605 ","End":"00:47.235","Text":"I\u0027d like to emphasize something important,"},{"Start":"00:47.235 ","End":"00:50.480","Text":"and that is those boxes that I\u0027ve highlighted them,"},{"Start":"00:50.480 ","End":"00:52.085","Text":"they all have to be the same."},{"Start":"00:52.085 ","End":"00:54.005","Text":"Here they are not."},{"Start":"00:54.005 ","End":"01:02.795","Text":"We have the x here and we have x here but what we have here is not x, it\u0027s sine x."},{"Start":"01:02.795 ","End":"01:07.760","Text":"Let\u0027s use some algebra and maybe a bit of trigonometry."},{"Start":"01:07.760 ","End":"01:12.485","Text":"What we have here is we have the limit as x"},{"Start":"01:12.485 ","End":"01:18.775","Text":"goes to 0 of 1 plus sine x."},{"Start":"01:18.775 ","End":"01:25.055","Text":"What I would like to have here is 1 over sine x,"},{"Start":"01:25.055 ","End":"01:26.630","Text":"but that\u0027s not what I have."},{"Start":"01:26.630 ","End":"01:28.160","Text":"I have 1 over x."},{"Start":"01:28.160 ","End":"01:32.690","Text":"Let\u0027s compensate by put a sine x in the denominator,"},{"Start":"01:32.690 ","End":"01:35.945","Text":"we need to put a sine x in the numerator,"},{"Start":"01:35.945 ","End":"01:37.495","Text":"so it cancels out."},{"Start":"01:37.495 ","End":"01:39.735","Text":"But we still want to have 1 over x,"},{"Start":"01:39.735 ","End":"01:41.405","Text":"so I\u0027ll put x here."},{"Start":"01:41.405 ","End":"01:42.740","Text":"If you multiply this out,"},{"Start":"01:42.740 ","End":"01:44.410","Text":"you get exactly 1 over x,"},{"Start":"01:44.410 ","End":"01:47.015","Text":"I\u0027ll also put brackets around here."},{"Start":"01:47.015 ","End":"01:50.375","Text":"What I\u0027m implicitly using is the formula"},{"Start":"01:50.375 ","End":"01:58.200","Text":"that (a^b)^c is a^bc."},{"Start":"01:58.200 ","End":"01:59.715","Text":"You\u0027ve seen this stuff before."},{"Start":"01:59.715 ","End":"02:02.340","Text":"Now the thing is, what about this?"},{"Start":"02:02.340 ","End":"02:04.235","Text":"If x goes to 0,"},{"Start":"02:04.235 ","End":"02:08.915","Text":"then sine x also goes to 0 because sine of 0 is 0."},{"Start":"02:08.915 ","End":"02:12.570","Text":"In other words, I have here sine x,"},{"Start":"02:12.570 ","End":"02:14.715","Text":"and I have here sine x,"},{"Start":"02:14.715 ","End":"02:17.100","Text":"and I have here sine x."},{"Start":"02:17.100 ","End":"02:18.350","Text":"This looks good."},{"Start":"02:18.350 ","End":"02:21.725","Text":"There is another formula that we\u0027re going to use,"},{"Start":"02:21.725 ","End":"02:31.445","Text":"and that is that the limit as x goes to 0 of sine x over x is equal to 1."},{"Start":"02:31.445 ","End":"02:33.050","Text":"Now we\u0027re almost done,"},{"Start":"02:33.050 ","End":"02:40.974","Text":"because this limit of 1 plus sine x^1 over sine x is exactly this formula."},{"Start":"02:40.974 ","End":"02:43.530","Text":"This part gives us e,"},{"Start":"02:43.530 ","End":"02:47.300","Text":"and here if we take the limit of the sine x over x,"},{"Start":"02:47.300 ","End":"02:49.044","Text":"we use this formula."},{"Start":"02:49.044 ","End":"02:51.705","Text":"This part gives us the 1."},{"Start":"02:51.705 ","End":"02:54.705","Text":"We get e^1 which is just e,"},{"Start":"02:54.705 ","End":"02:56.830","Text":"and that\u0027s the answer."}],"ID":4786},{"Watched":false,"Name":"Exercise 7","Duration":"5m 2s","ChapterTopicVideoID":4779,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.075","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression,"},{"Start":"00:06.075 ","End":"00:09.030","Text":"which is something to the power of x."},{"Start":"00:09.030 ","End":"00:13.020","Text":"If we just take that something is 2x plus 3 over 2x minus 3,"},{"Start":"00:13.020 ","End":"00:14.910","Text":"and use our usual tricks,"},{"Start":"00:14.910 ","End":"00:19.425","Text":"we see that this tends to 1 and the x goes to infinity,"},{"Start":"00:19.425 ","End":"00:20.940","Text":"so what we end up with,"},{"Start":"00:20.940 ","End":"00:23.730","Text":"1 to the power of infinity."},{"Start":"00:23.730 ","End":"00:28.625","Text":"That\u0027s one of those indeterminate undefined expressions."},{"Start":"00:28.625 ","End":"00:32.300","Text":"What we have to do is use some trick which is"},{"Start":"00:32.300 ","End":"00:36.185","Text":"probably going to be Euler\u0027s formula when we have 1 to the infinity."},{"Start":"00:36.185 ","End":"00:38.480","Text":"Anyway, let\u0027s see what Euler\u0027s limit is."},{"Start":"00:38.480 ","End":"00:42.450","Text":"Now, we have to use this in our case."},{"Start":"00:42.450 ","End":"00:48.665","Text":"What we have inside the brackets doesn\u0027t look at all like 1 plus 1 over something."},{"Start":"00:48.665 ","End":"00:50.505","Text":"We have this thing."},{"Start":"00:50.505 ","End":"00:54.650","Text":"What we\u0027re going to do is use a bit of algebra and twist it a bit,"},{"Start":"00:54.650 ","End":"00:57.740","Text":"spoil it, then fix it until we get it like that."},{"Start":"00:57.740 ","End":"00:59.690","Text":"Well, I\u0027ll just take the inside,"},{"Start":"00:59.690 ","End":"01:09.710","Text":"this part here and say that 2x plus 3 over 2x minus 3 is going to equal,"},{"Start":"01:09.710 ","End":"01:14.640","Text":"make it in the form 1 over 1 plus 1 over the box."},{"Start":"01:14.640 ","End":"01:17.615","Text":"In other words, if I do a bit of algebra,"},{"Start":"01:17.615 ","End":"01:26.605","Text":"I\u0027m going to bring the 1 over to the other side and I\u0027ll get is equal to 1 over box."},{"Start":"01:26.605 ","End":"01:30.170","Text":"Therefore, if I put a common denominator,"},{"Start":"01:30.170 ","End":"01:34.865","Text":"I get and this equals the same 1 over box."},{"Start":"01:34.865 ","End":"01:37.130","Text":"Now, simplify the numerator,"},{"Start":"01:37.130 ","End":"01:47.385","Text":"do the algebra and we get 6 over 2x minus 3 is 1 over this box."},{"Start":"01:47.385 ","End":"01:53.320","Text":"If we take the reciprocal of each side 1 over and switch sides as well,"},{"Start":"01:53.320 ","End":"02:00.950","Text":"that our box is equal to 2x minus 3 over 6."},{"Start":"02:00.950 ","End":"02:07.195","Text":"Again, scroll down a bit and our original limit becomes this."},{"Start":"02:07.195 ","End":"02:10.180","Text":"This was a simplification of what was this"},{"Start":"02:10.180 ","End":"02:13.970","Text":"over this to the power of x and I simplify the inside."},{"Start":"02:13.970 ","End":"02:15.985","Text":"Going back down again,"},{"Start":"02:15.985 ","End":"02:22.030","Text":"what we had was that same thing that was inside simplifies to this and we still have x."},{"Start":"02:22.030 ","End":"02:29.785","Text":"However, this still doesn\u0027t look very much like 1 plus 1 over box to the power of box,"},{"Start":"02:29.785 ","End":"02:32.175","Text":"which is what e is,"},{"Start":"02:32.175 ","End":"02:34.590","Text":"so let\u0027s do a bit more work."},{"Start":"02:34.590 ","End":"02:37.350","Text":"This is equal to the limit,"},{"Start":"02:37.350 ","End":"02:40.510","Text":"I\u0027ll leave for a second what\u0027s below the limit,"},{"Start":"02:40.510 ","End":"02:47.355","Text":"1 plus 1 over 2x minus 3 over 6."},{"Start":"02:47.355 ","End":"02:49.530","Text":"Here I have x."},{"Start":"02:49.530 ","End":"02:55.305","Text":"What I want here is the limit of 1 plus 1 over"},{"Start":"02:55.305 ","End":"03:03.060","Text":"2x minus 3 over 6 to the power of what this box is,"},{"Start":"03:03.060 ","End":"03:06.410","Text":"which is 2x minus 3 over 6."},{"Start":"03:06.410 ","End":"03:08.860","Text":"But I\u0027ve changed the exercise,"},{"Start":"03:08.860 ","End":"03:15.335","Text":"so I have to multiply this back by something that gives me x."},{"Start":"03:15.335 ","End":"03:18.110","Text":"What I can do is, first of all,"},{"Start":"03:18.110 ","End":"03:23.285","Text":"multiply by the reciprocal and that brings it back to 1."},{"Start":"03:23.285 ","End":"03:28.665","Text":"It\u0027s 6 over 2x minus 3."},{"Start":"03:28.665 ","End":"03:31.175","Text":"Now it\u0027s 1, but I need it to be x,"},{"Start":"03:31.175 ","End":"03:33.545","Text":"so I\u0027ll put x in the numerator here."},{"Start":"03:33.545 ","End":"03:37.040","Text":"Also, I\u0027m going to put a square bracket around here because"},{"Start":"03:37.040 ","End":"03:40.530","Text":"the product of powers is this to the power of this,"},{"Start":"03:40.530 ","End":"03:42.195","Text":"and this is what we have."},{"Start":"03:42.195 ","End":"03:45.540","Text":"We have the 2x minus 3 here,"},{"Start":"03:45.540 ","End":"03:48.130","Text":"we have the 2x minus 3 here."},{"Start":"03:48.130 ","End":"03:54.765","Text":"But the limit originally was x goes to infinity,"},{"Start":"03:54.765 ","End":"04:00.080","Text":"but what we have is not x but the box which is 2x minus 3 over 6."},{"Start":"04:00.080 ","End":"04:02.630","Text":"But when this goes to infinity,"},{"Start":"04:02.630 ","End":"04:09.690","Text":"it\u0027s the same thing as 2x minus 3 over 6 goes to infinity."},{"Start":"04:09.690 ","End":"04:16.770","Text":"We could have replaced this x by the box and that makes no difference."},{"Start":"04:16.770 ","End":"04:20.960","Text":"Here, here, and here we have the same thing."},{"Start":"04:20.960 ","End":"04:23.300","Text":"Now that we know that this goes to e,"},{"Start":"04:23.300 ","End":"04:28.729","Text":"what I can do is say that this is equal to the limit"},{"Start":"04:28.729 ","End":"04:35.625","Text":"of e to the power of 6x over 2x minus 3,"},{"Start":"04:35.625 ","End":"04:37.680","Text":"because this thing is like here,"},{"Start":"04:37.680 ","End":"04:39.345","Text":"it\u0027s e. Now,"},{"Start":"04:39.345 ","End":"04:40.800","Text":"the limit of this,"},{"Start":"04:40.800 ","End":"04:44.265","Text":"of 6x over 2x plus 3,"},{"Start":"04:44.265 ","End":"04:46.370","Text":"we\u0027ve done this trick many times,"},{"Start":"04:46.370 ","End":"04:48.560","Text":"taking x outside the brackets,"},{"Start":"04:48.560 ","End":"04:52.420","Text":"it just comes out to be 6 over 2, which is 3."},{"Start":"04:52.420 ","End":"04:58.700","Text":"This is just equal to e to the power of 3 and in fact,"},{"Start":"04:58.700 ","End":"05:02.970","Text":"we finally got to the answer, and that\u0027s it."}],"ID":4787},{"Watched":false,"Name":"Exercise 8","Duration":"9m 40s","ChapterTopicVideoID":4780,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we have to find the limit as"},{"Start":"00:03.480 ","End":"00:07.290","Text":"x goes to infinity of this whole expression."},{"Start":"00:07.290 ","End":"00:10.020","Text":"The first thing to do is to try substituting x"},{"Start":"00:10.020 ","End":"00:13.105","Text":"equals infinity and see if something goes wrong."},{"Start":"00:13.105 ","End":"00:17.900","Text":"Well, if we put x equals infinity inside the brackets, we get 1."},{"Start":"00:17.900 ","End":"00:20.960","Text":"Using our usual technique of taking out the highest power,"},{"Start":"00:20.960 ","End":"00:24.810","Text":"we take out x squared and we get 1 plus 1/x, etc."},{"Start":"00:24.810 ","End":"00:26.620","Text":"You\u0027ll see that you get 1."},{"Start":"00:26.620 ","End":"00:28.580","Text":"Here, x goes to infinity,"},{"Start":"00:28.580 ","End":"00:30.140","Text":"this whole thing goes to infinity."},{"Start":"00:30.140 ","End":"00:33.995","Text":"This is of the form 1 to the power of infinity."},{"Start":"00:33.995 ","End":"00:36.110","Text":"When you get 1 to the power of infinity,"},{"Start":"00:36.110 ","End":"00:41.480","Text":"it\u0027s undefined and often indicates that we should be using Euler\u0027s limit."},{"Start":"00:41.480 ","End":"00:44.720","Text":"I\u0027ve already pre-written Euler\u0027s limit to save time,"},{"Start":"00:44.720 ","End":"00:47.555","Text":"at the limit of something."},{"Start":"00:47.555 ","End":"00:50.930","Text":"It doesn\u0027t have to be x, but some expression goes to"},{"Start":"00:50.930 ","End":"00:54.275","Text":"infinity of 1 plus 1 over this something, let\u0027s call it box,"},{"Start":"00:54.275 ","End":"00:56.740","Text":"to the power of box is equal to"},{"Start":"00:56.740 ","End":"01:01.280","Text":"e. What we have here doesn\u0027t really look very much like this."},{"Start":"01:01.280 ","End":"01:05.615","Text":"We\u0027re going to somehow force it to get more into shape like this."},{"Start":"01:05.615 ","End":"01:10.640","Text":"What I\u0027d like to do is to start off with the bit inside the brackets,"},{"Start":"01:10.640 ","End":"01:16.250","Text":"this bit, and try to get it to look like 1 plus 1 over box."},{"Start":"01:16.250 ","End":"01:20.420","Text":"In other words, what I want to do is to get this to look something"},{"Start":"01:20.420 ","End":"01:25.630","Text":"like 1 plus 1 over box. Let\u0027s see."},{"Start":"01:25.630 ","End":"01:27.410","Text":"I\u0027ll continue at the side here."},{"Start":"01:27.410 ","End":"01:29.090","Text":"If this is what I want it to be,"},{"Start":"01:29.090 ","End":"01:36.830","Text":"then we get the expression that 1 plus 1 over box is equal"},{"Start":"01:36.830 ","End":"01:45.765","Text":"to x squared plus x plus 1/x squared plus x plus 4."},{"Start":"01:45.765 ","End":"01:51.055","Text":"We want the 1 over box that it\u0027s equal to this thing minus 1."},{"Start":"01:51.055 ","End":"01:55.875","Text":"This is equal to x squared plus x plus"},{"Start":"01:55.875 ","End":"02:01.950","Text":"1/x squared plus x plus 4 minus 1."},{"Start":"02:01.950 ","End":"02:05.935","Text":"Now let\u0027s put the right-hand side over a common denominator."},{"Start":"02:05.935 ","End":"02:10.310","Text":"What we get is common denominator will be x"},{"Start":"02:10.310 ","End":"02:14.630","Text":"squared plus x plus 4 of the denominator here."},{"Start":"02:14.630 ","End":"02:20.780","Text":"The numerator will be x squared plus x plus 1,"},{"Start":"02:20.780 ","End":"02:21.890","Text":"just as it was."},{"Start":"02:21.890 ","End":"02:26.614","Text":"But the minus 1, I can write as this thing over itself."},{"Start":"02:26.614 ","End":"02:28.340","Text":"A over a is always 1."},{"Start":"02:28.340 ","End":"02:30.350","Text":"If I have this in the denominator,"},{"Start":"02:30.350 ","End":"02:32.705","Text":"I also have to have it in the numerator,"},{"Start":"02:32.705 ","End":"02:34.705","Text":"but it\u0027s a minus 1."},{"Start":"02:34.705 ","End":"02:41.740","Text":"I\u0027m going to put it in brackets and it will be x squared plus x plus 4."},{"Start":"02:44.710 ","End":"02:48.470","Text":"This is equal to, well, let\u0027s see,"},{"Start":"02:48.470 ","End":"02:54.350","Text":"x squared cancels out with x squared with the minus,"},{"Start":"02:54.350 ","End":"02:58.264","Text":"the x and the minus x cancel,"},{"Start":"02:58.264 ","End":"03:01.010","Text":"and all we\u0027re left with is 1 minus 4,"},{"Start":"03:01.010 ","End":"03:02.959","Text":"which is equal to 3."},{"Start":"03:02.959 ","End":"03:10.735","Text":"What we have is minus 3/x squared plus x plus 4."},{"Start":"03:10.735 ","End":"03:14.735","Text":"All this is 1 over box."},{"Start":"03:14.735 ","End":"03:17.570","Text":"Now this is easy because if we want to find box,"},{"Start":"03:17.570 ","End":"03:20.660","Text":"all we have to do is take the reciprocal on both sides,"},{"Start":"03:20.660 ","End":"03:26.000","Text":"invert the numerator and the denominator and we get that box is equal to"},{"Start":"03:26.000 ","End":"03:32.605","Text":"x squared plus x plus 4 over minus 3."},{"Start":"03:32.605 ","End":"03:38.870","Text":"Now, I\u0027m going to look back at what the original exercise was."},{"Start":"03:38.870 ","End":"03:43.550","Text":"What I\u0027m going to do now that we\u0027ve found what the square is, the box,"},{"Start":"03:43.550 ","End":"03:51.920","Text":"I\u0027m just going to put it back in here and so we have limit of 1 plus 1 over box."},{"Start":"03:51.920 ","End":"03:54.365","Text":"I\u0027ll copy this over here."},{"Start":"03:54.365 ","End":"03:58.475","Text":"X squared plus x plus 4"},{"Start":"03:58.475 ","End":"04:05.310","Text":"over minus 3 as x goes to infinity."},{"Start":"04:05.310 ","End":"04:09.645","Text":"All this was to the power of 4x squared."},{"Start":"04:09.645 ","End":"04:12.050","Text":"Now we have 1 plus 1 over something,"},{"Start":"04:12.050 ","End":"04:18.770","Text":"but this something in the denominator here is not the same as what\u0027s up here."},{"Start":"04:18.770 ","End":"04:22.400","Text":"In other words, this yellow and this yellow are just not the same."},{"Start":"04:22.400 ","End":"04:25.880","Text":"What I automatically do in situations like"},{"Start":"04:25.880 ","End":"04:31.025","Text":"this is you could call it the spoil and repair. I\u0027ll show you what I mean."},{"Start":"04:31.025 ","End":"04:36.350","Text":"I mean that what I want if x goes to infinity,"},{"Start":"04:36.350 ","End":"04:46.755","Text":"1 plus 1/x squared plus x plus 4 over minus 3."},{"Start":"04:46.755 ","End":"04:50.780","Text":"What I really want up here is not the 4x squared."},{"Start":"04:50.780 ","End":"04:58.925","Text":"I would really like to have x squared plus x plus 4 over minus 3."},{"Start":"04:58.925 ","End":"05:02.780","Text":"But hey, I can\u0027t just change the exercise in the middle."},{"Start":"05:02.780 ","End":"05:05.915","Text":"This is what the exercise was and I suddenly changed it to this."},{"Start":"05:05.915 ","End":"05:09.725","Text":"That\u0027s what I call the spoiling it part and I\u0027m going to repair it."},{"Start":"05:09.725 ","End":"05:11.885","Text":"If I want it to be 4x squared,"},{"Start":"05:11.885 ","End":"05:16.595","Text":"I could do an algebra problem at the side and say this times what is this?"},{"Start":"05:16.595 ","End":"05:20.080","Text":"But I find it easier to just correct."},{"Start":"05:20.080 ","End":"05:24.500","Text":"If I put an x squared plus x plus 4 in the numerator,"},{"Start":"05:24.500 ","End":"05:27.650","Text":"and I also put it in the denominator,"},{"Start":"05:27.650 ","End":"05:30.780","Text":"x squared plus x plus 4,"},{"Start":"05:30.780 ","End":"05:32.670","Text":"then this cancels out."},{"Start":"05:32.670 ","End":"05:37.610","Text":"If I put this in the numerator minus 3, this cancels out."},{"Start":"05:37.610 ","End":"05:43.460","Text":"But we still have to have the 4x squared times 4x squared."},{"Start":"05:43.460 ","End":"05:48.965","Text":"Now, I want to just simplify what\u0027s in the numerator here."},{"Start":"05:48.965 ","End":"05:53.075","Text":"We can do that at the side here and just copy."},{"Start":"05:53.075 ","End":"05:57.590","Text":"All I want to simplify is this bit here,"},{"Start":"05:57.590 ","End":"06:01.715","Text":"because this bit goes with this where we highlighted in yellow."},{"Start":"06:01.715 ","End":"06:09.665","Text":"What we get here is minus 3/x squared plus x plus 4 times"},{"Start":"06:09.665 ","End":"06:19.050","Text":"4x squared is just equal to minus 12x squared over x squared plus x plus 4."},{"Start":"06:19.050 ","End":"06:24.170","Text":"Our usual trick of taking the highest power outside the brackets."},{"Start":"06:24.170 ","End":"06:26.450","Text":"Well, that\u0027s good for us here too."},{"Start":"06:26.450 ","End":"06:32.260","Text":"This equals x squared and what we\u0027re left with is minus 12."},{"Start":"06:32.260 ","End":"06:36.860","Text":"Over here, x squared outside the brackets leaves us with"},{"Start":"06:36.860 ","End":"06:45.850","Text":"1 plus 1/x plus 4/x squared and the x squared cancel."},{"Start":"06:45.850 ","End":"06:48.545","Text":"Now going back to here,"},{"Start":"06:48.545 ","End":"06:53.885","Text":"what I get is the limit as x goes to infinity of"},{"Start":"06:53.885 ","End":"07:00.265","Text":"1 plus 1/x squared plus x plus 4,"},{"Start":"07:00.265 ","End":"07:06.960","Text":"the same x squared plus x plus 4 over minus 3."},{"Start":"07:06.960 ","End":"07:13.040","Text":"What I can do is put this in a square brackets and outside here put what this was,"},{"Start":"07:13.040 ","End":"07:19.400","Text":"which is minus 12/1 plus"},{"Start":"07:19.400 ","End":"07:24.010","Text":"1/x plus 4 over x squared."},{"Start":"07:24.010 ","End":"07:31.160","Text":"What I\u0027m really doing here is using the basic algebraic formula that in general,"},{"Start":"07:31.160 ","End":"07:38.510","Text":"a to the power of b to the power of c is a to the power of b times c,"},{"Start":"07:38.510 ","End":"07:42.090","Text":"where this part is the b,"},{"Start":"07:42.090 ","End":"07:45.290","Text":"this part is the c. When we multiply them all together,"},{"Start":"07:45.290 ","End":"07:50.105","Text":"we get this, which is the a in that formula down there."},{"Start":"07:50.105 ","End":"07:52.550","Text":"Now we\u0027re all ready for the limit."},{"Start":"07:52.550 ","End":"07:54.590","Text":"Because when x goes to infinity,"},{"Start":"07:54.590 ","End":"07:57.575","Text":"1 plus box to the power of box,"},{"Start":"07:57.575 ","End":"08:01.340","Text":"this is the same as this,"},{"Start":"08:01.340 ","End":"08:04.880","Text":"which takes the place of what we had in the box here."},{"Start":"08:04.880 ","End":"08:12.095","Text":"There\u0027s just 1 slight thing that I should mention is that in the formula over here,"},{"Start":"08:12.095 ","End":"08:15.005","Text":"we say that box goes to infinity,"},{"Start":"08:15.005 ","End":"08:19.775","Text":"whereas here we have x goes to infinity."},{"Start":"08:19.775 ","End":"08:22.985","Text":"But when x goes to infinity,"},{"Start":"08:22.985 ","End":"08:26.135","Text":"box also goes to infinity,"},{"Start":"08:26.135 ","End":"08:28.700","Text":"because if x goes to infinity,"},{"Start":"08:28.700 ","End":"08:31.160","Text":"then x squared plus x plus 4,"},{"Start":"08:31.160 ","End":"08:33.070","Text":"also go to infinity."},{"Start":"08:33.070 ","End":"08:36.030","Text":"When we divide by minus 3,"},{"Start":"08:36.030 ","End":"08:38.815","Text":"it\u0027s not infinity, it\u0027s minus infinity."},{"Start":"08:38.815 ","End":"08:41.405","Text":"That goes to minus infinity."},{"Start":"08:41.405 ","End":"08:48.170","Text":"However, that\u0027s still good because this formula also holds for minus infinity."},{"Start":"08:48.170 ","End":"08:52.145","Text":"In other words, even if box goes to minus infinity,"},{"Start":"08:52.145 ","End":"08:54.745","Text":"this formula still holds true."},{"Start":"08:54.745 ","End":"08:58.010","Text":"What we get here is we look at this in 2 parts."},{"Start":"08:58.010 ","End":"08:59.030","Text":"We have the first part,"},{"Start":"08:59.030 ","End":"09:01.700","Text":"1 plus 1 over box to the power of box,"},{"Start":"09:01.700 ","End":"09:05.690","Text":"and that equals e. We\u0027ll just do that part first."},{"Start":"09:05.690 ","End":"09:07.070","Text":"We have the limit,"},{"Start":"09:07.070 ","End":"09:13.279","Text":"x goes to infinity of e to the power of minus 12 over"},{"Start":"09:13.279 ","End":"09:19.875","Text":"1 plus 1/x plus 4/x squared."},{"Start":"09:19.875 ","End":"09:23.920","Text":"In this case we can just substitute the x equals infinity."},{"Start":"09:23.920 ","End":"09:29.690","Text":"What we get is minus 12/1 plus 0 plus 0."},{"Start":"09:29.690 ","End":"09:32.210","Text":"That altogether comes out as minus 12."},{"Start":"09:32.210 ","End":"09:33.710","Text":"The denominator is 1."},{"Start":"09:33.710 ","End":"09:38.930","Text":"We just get that this is e to the power of minus 12."},{"Start":"09:38.930 ","End":"09:41.260","Text":"That\u0027s the answer."}],"ID":4788},{"Watched":false,"Name":"Exercise 9","Duration":"7m 49s","ChapterTopicVideoID":4781,"CourseChapterTopicPlaylistID":3701,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.285","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression."},{"Start":"00:06.285 ","End":"00:11.610","Text":"If we just substitute x equals infinity, we\u0027ll get 1^infinity."},{"Start":"00:11.610 ","End":"00:15.284","Text":"This is because 1/x goes to 0,"},{"Start":"00:15.284 ","End":"00:17.370","Text":"tangent to 0 is 0 and so on."},{"Start":"00:17.370 ","End":"00:19.680","Text":"We\u0027re left with 1^infinity and that\u0027s"},{"Start":"00:19.680 ","End":"00:24.420","Text":"undefined and often indicates that we should be using Euler\u0027s formula."},{"Start":"00:24.420 ","End":"00:29.745","Text":"I pre-wrote it already and this is what Euler\u0027s formula says."},{"Start":"00:29.745 ","End":"00:32.160","Text":"When something goes to infinity,"},{"Start":"00:32.160 ","End":"00:36.930","Text":"let\u0027s call it box of 1 plus 1/box all to the power of box that gives"},{"Start":"00:36.930 ","End":"00:41.995","Text":"us E. But these 3 things I\u0027ve highlighted should be the same quantity."},{"Start":"00:41.995 ","End":"00:46.580","Text":"The main differences is that here we have 1 over something and here we don\u0027t,"},{"Start":"00:46.580 ","End":"00:50.360","Text":"we just have the tangent that we do have a 1 over inside."},{"Start":"00:50.360 ","End":"00:57.260","Text":"Let\u0027s see if we can do some algebra and trigonometry and get it more into this form."},{"Start":"00:57.260 ","End":"01:00.440","Text":"Now, the first thing I can do,"},{"Start":"01:00.440 ","End":"01:03.080","Text":"I want to write it as 1 over something."},{"Start":"01:03.080 ","End":"01:07.015","Text":"So I\u0027ll use the fact that 1/tangent,"},{"Start":"01:07.015 ","End":"01:08.820","Text":"doesn\u0027t matter of what tangent x,"},{"Start":"01:08.820 ","End":"01:11.660","Text":"tangent Alpha, 1/tangent is cotangent."},{"Start":"01:11.660 ","End":"01:13.220","Text":"This is just symbolically."},{"Start":"01:13.220 ","End":"01:20.645","Text":"The reason is, is that the tangent is the sine over the cosine of whatever it is."},{"Start":"01:20.645 ","End":"01:27.260","Text":"The cotangent is the cosine over the sine of whatever it is,"},{"Start":"01:27.260 ","End":"01:28.910","Text":"x or Alpha whatever."},{"Start":"01:28.910 ","End":"01:34.330","Text":"Clearly, each one of these is 1 over the other."},{"Start":"01:34.330 ","End":"01:38.165","Text":"More than that tangent times cotangent is 1,"},{"Start":"01:38.165 ","End":"01:39.725","Text":"might be useful also."},{"Start":"01:39.725 ","End":"01:44.090","Text":"Of course, we\u0027ll also probably be using the algebraic formula"},{"Start":"01:44.090 ","End":"01:51.030","Text":"that a^b^c is going to be a^bc."},{"Start":"01:51.030 ","End":"01:55.835","Text":"What we\u0027ll do is we\u0027ll write the tangent as 1/cotangent."},{"Start":"01:55.835 ","End":"01:59.620","Text":"Tangent is 1/cotangent."},{"Start":"01:59.620 ","End":"02:06.825","Text":"Here we\u0027ll get the limit as x goes to infinity of 1"},{"Start":"02:06.825 ","End":"02:15.650","Text":"plus 1/cotangent of 1/x and all this to the power of x."},{"Start":"02:15.650 ","End":"02:17.300","Text":"Now we\u0027ve got the 1 over,"},{"Start":"02:17.300 ","End":"02:20.480","Text":"but we need to make sure that these two are the same."},{"Start":"02:20.480 ","End":"02:30.200","Text":"What I\u0027ll do is I\u0027ll first of all force it in a way, so 1/cotangent 1/x."},{"Start":"02:30.200 ","End":"02:36.040","Text":"Instead of x I\u0027ll force it to be cotangent of 1/x."},{"Start":"02:36.040 ","End":"02:38.840","Text":"I\u0027ll still need the limit."},{"Start":"02:38.840 ","End":"02:42.440","Text":"Obviously, I can\u0027t just force a change in the problem."},{"Start":"02:42.440 ","End":"02:45.365","Text":"If I\u0027ve broken it, I have to fix it."},{"Start":"02:45.365 ","End":"02:53.515","Text":"What I\u0027ll do is write this as the limit x goes to infinity of"},{"Start":"02:53.515 ","End":"03:02.505","Text":"1 plus 1/cotangent 1/x^cotangent 1/x."},{"Start":"03:02.505 ","End":"03:05.670","Text":"This is still the broken expression."},{"Start":"03:05.670 ","End":"03:07.650","Text":"This is where it\u0027s wrong,"},{"Start":"03:07.650 ","End":"03:09.830","Text":"as I said that this is the thing that has to be fixed."},{"Start":"03:09.830 ","End":"03:11.810","Text":"This is not a line in the solution."},{"Start":"03:11.810 ","End":"03:14.300","Text":"It\u0027s just what I wish it was."},{"Start":"03:14.300 ","End":"03:18.815","Text":"But I can make my wish come true if I compensate."},{"Start":"03:18.815 ","End":"03:21.380","Text":"To make this x here,"},{"Start":"03:21.380 ","End":"03:28.835","Text":"if I multiply by tangent 1/x and cotangent times tangent,"},{"Start":"03:28.835 ","End":"03:31.220","Text":"as I mentioned before, this times this is equal"},{"Start":"03:31.220 ","End":"03:33.755","Text":"to 1 as each is a reciprocal is of the other."},{"Start":"03:33.755 ","End":"03:36.550","Text":"Now we have 1, but what we had was x,"},{"Start":"03:36.550 ","End":"03:40.040","Text":"so if I also put an x here and I\u0027ve fixed it."},{"Start":"03:40.040 ","End":"03:45.405","Text":"Of course there\u0027s another way of doing this and that\u0027s by doing an equation that"},{"Start":"03:45.405 ","End":"03:49.230","Text":"cotangent of 1/x times some letter equals"},{"Start":"03:49.230 ","End":"03:52.565","Text":"x and solving for that letter and getting it is equal to this."},{"Start":"03:52.565 ","End":"04:00.905","Text":"You could. The first thing we can do is do the limit of the inside part using this,"},{"Start":"04:00.905 ","End":"04:05.075","Text":"where this square is cotangent 1/x."},{"Start":"04:05.075 ","End":"04:09.920","Text":"Let\u0027s just notice that when x goes to infinity,"},{"Start":"04:09.920 ","End":"04:14.180","Text":"also cotangent of 1/x goes to infinity and vice versa."},{"Start":"04:14.180 ","End":"04:17.735","Text":"Because if x goes to infinity, 1/x is 0,"},{"Start":"04:17.735 ","End":"04:21.855","Text":"cotangent of 0 is infinity,"},{"Start":"04:21.855 ","End":"04:27.080","Text":"so this thing actually also goes to infinity."},{"Start":"04:27.080 ","End":"04:28.865","Text":"When x goes to infinity,"},{"Start":"04:28.865 ","End":"04:35.705","Text":"that cotangent of 1/x also goes to infinity."},{"Start":"04:35.705 ","End":"04:39.795","Text":"In fact, in the same way that if x goes to plus infinity,"},{"Start":"04:39.795 ","End":"04:45.515","Text":"1/x goes to plus 0 and this thing goes to plus infinity and minus and minus."},{"Start":"04:45.515 ","End":"04:47.675","Text":"All these three work,"},{"Start":"04:47.675 ","End":"04:49.790","Text":"this yellow, this yellow, and this yellow."},{"Start":"04:49.790 ","End":"04:51.470","Text":"Here we have the cotangent 1/x,"},{"Start":"04:51.470 ","End":"04:54.725","Text":"here it is and it also goes to infinity."},{"Start":"04:54.725 ","End":"04:58.370","Text":"All this thing now justifies that we write that"},{"Start":"04:58.370 ","End":"05:02.990","Text":"all this part goes to e. We still have the limit to do though,"},{"Start":"05:02.990 ","End":"05:05.630","Text":"because we haven\u0027t gotten rid of the x altogether."},{"Start":"05:05.630 ","End":"05:14.400","Text":"This part, x goes to infinity of e^x times tangent 1/x."},{"Start":"05:14.400 ","End":"05:19.685","Text":"I\u0027ll do this also at the side and do a bit of trigonometry and algebra."},{"Start":"05:19.685 ","End":"05:24.500","Text":"X times tangent 1/x is"},{"Start":"05:24.500 ","End":"05:31.130","Text":"equal to tangent 1/x/1/x."},{"Start":"05:31.130 ","End":"05:34.010","Text":"If I put the number from the numerator in the denominator,"},{"Start":"05:34.010 ","End":"05:36.560","Text":"the fraction reverses or the other way around."},{"Start":"05:36.560 ","End":"05:38.135","Text":"If I put 1/x here,"},{"Start":"05:38.135 ","End":"05:40.220","Text":"it\u0027s like putting x in the numerator."},{"Start":"05:40.220 ","End":"05:43.940","Text":"Dividing by a fraction is like multiplying by its inverse."},{"Start":"05:43.940 ","End":"05:48.530","Text":"Now I\u0027d rather have sine than tangent because I\u0027m thinking to myself,"},{"Start":"05:48.530 ","End":"05:53.660","Text":"x goes to 0 and tangent of something over something when that something goes to 0,"},{"Start":"05:53.660 ","End":"05:56.540","Text":"I do know, so I\u0027d prefer a sine than a tangent."},{"Start":"05:56.540 ","End":"06:04.820","Text":"So I\u0027ll write this as sine of 1/x/1/x."},{"Start":"06:04.820 ","End":"06:08.885","Text":"But again, I can\u0027t just change the exercise."},{"Start":"06:08.885 ","End":"06:14.100","Text":"But if you remember that tangent is sine/cosine, then here,"},{"Start":"06:14.100 ","End":"06:19.845","Text":"I can just multiply it by 1/cosine."},{"Start":"06:19.845 ","End":"06:23.090","Text":"This times this. This was this is the tangent and here\u0027s the"},{"Start":"06:23.090 ","End":"06:26.770","Text":"1/x that we had and it all works out."},{"Start":"06:26.770 ","End":"06:30.965","Text":"At this point, I\u0027ll need to do another trick"},{"Start":"06:30.965 ","End":"06:35.345","Text":"which you may or may not be familiar with and that is a trick of substitution."},{"Start":"06:35.345 ","End":"06:42.245","Text":"What we do is we basically put t in place of 1/x."},{"Start":"06:42.245 ","End":"06:46.790","Text":"Notice that when x goes to infinity,"},{"Start":"06:46.790 ","End":"06:50.420","Text":"that t goes to 0."},{"Start":"06:50.420 ","End":"06:53.840","Text":"Actually 0 plus, but it won\u0027t make any difference."},{"Start":"06:53.840 ","End":"06:59.705","Text":"What we get is that this thing is equal to sine"},{"Start":"06:59.705 ","End":"07:06.395","Text":"of t/t times 1/cosine t,"},{"Start":"07:06.395 ","End":"07:08.380","Text":"and t goes to 0."},{"Start":"07:08.380 ","End":"07:11.010","Text":"The answer to this is what goes here."},{"Start":"07:11.010 ","End":"07:14.220","Text":"Let me first of all write the limit x"},{"Start":"07:14.220 ","End":"07:18.755","Text":"goes to infinity of e to the power of and we\u0027ll see."},{"Start":"07:18.755 ","End":"07:21.800","Text":"Now this thing, when t goes to 0,"},{"Start":"07:21.800 ","End":"07:25.685","Text":"sine t/t is a famous limit and that goes to 1."},{"Start":"07:25.685 ","End":"07:29.165","Text":"Cosine t, when you put t equals 0,"},{"Start":"07:29.165 ","End":"07:30.410","Text":"we don\u0027t have to do a limit."},{"Start":"07:30.410 ","End":"07:34.205","Text":"We can just substitute cosine of 0 is just 1."},{"Start":"07:34.205 ","End":"07:36.560","Text":"This also goes to 1."},{"Start":"07:36.560 ","End":"07:39.830","Text":"1 times 1 is 1."},{"Start":"07:39.830 ","End":"07:42.145","Text":"This 1 times 1 goes here."},{"Start":"07:42.145 ","End":"07:49.450","Text":"This is e to the power of 1 and that\u0027s just equal to e and that\u0027s our answer."}],"ID":4789}],"Thumbnail":null,"ID":3701},{"Name":"Technique 7 Trigonometric Limits","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Trigonometric Limits Part 1","Duration":"8m 8s","ChapterTopicVideoID":9305,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.795","Text":"In this clip, I\u0027m going to introduce trigonometric limits,"},{"Start":"00:03.795 ","End":"00:10.755","Text":"which is actually technique Number 7 in the series of how to solve limits of functions."},{"Start":"00:10.755 ","End":"00:12.120","Text":"The first question is,"},{"Start":"00:12.120 ","End":"00:14.550","Text":"what is a trigonometric limit?"},{"Start":"00:14.550 ","End":"00:16.680","Text":"In those general sense,"},{"Start":"00:16.680 ","End":"00:22.980","Text":"it\u0027s just the limit of a function which contains trigonometric functions within it,"},{"Start":"00:22.980 ","End":"00:26.295","Text":"though it\u0027s usually used in more specific contexts,"},{"Start":"00:26.295 ","End":"00:28.830","Text":"but that\u0027s the general definition."},{"Start":"00:28.830 ","End":"00:30.975","Text":"I\u0027ll give you few examples."},{"Start":"00:30.975 ","End":"00:39.885","Text":"For example, limit as x goes to 0 of sine x over 4x."},{"Start":"00:39.885 ","End":"00:42.260","Text":"That\u0027s a trigonometric limit."},{"Start":"00:42.260 ","End":"00:47.185","Text":"Another example, the limit as x goes to"},{"Start":"00:47.185 ","End":"00:54.675","Text":"Pi of 1 minus cosine x over x squared."},{"Start":"00:54.675 ","End":"00:57.900","Text":"Another example, limit, again,"},{"Start":"00:57.900 ","End":"01:00.075","Text":"x goes to 0, that\u0027s very common,"},{"Start":"01:00.075 ","End":"01:02.570","Text":"of something more complicated,"},{"Start":"01:02.570 ","End":"01:08.315","Text":"tangent x times sine x over"},{"Start":"01:08.315 ","End":"01:15.330","Text":"the square root of sine x times x."},{"Start":"01:15.330 ","End":"01:20.030","Text":"Just about any expression containing trigonometric functions."},{"Start":"01:20.030 ","End":"01:23.045","Text":"The main tool we have in helping us to solve"},{"Start":"01:23.045 ","End":"01:27.470","Text":"trigonometric limits is a theorem, and I\u0027ll write it."},{"Start":"01:27.470 ","End":"01:31.295","Text":"I don\u0027t even remember which mathematician discovered it,"},{"Start":"01:31.295 ","End":"01:33.305","Text":"but it says the following."},{"Start":"01:33.305 ","End":"01:42.380","Text":"That the limit as x goes to 0 of sine x over x is equal to 1."},{"Start":"01:42.380 ","End":"01:47.480","Text":"This is the main tool which we will use for solving trigonometric limits."},{"Start":"01:47.480 ","End":"01:48.710","Text":"It won\u0027t always help."},{"Start":"01:48.710 ","End":"01:51.409","Text":"Not all tools are applicable to all situations,"},{"Start":"01:51.409 ","End":"01:53.659","Text":"but this is very important."},{"Start":"01:53.659 ","End":"01:55.309","Text":"It\u0027s certainly not trivial."},{"Start":"01:55.309 ","End":"01:58.250","Text":"I mean, you can\u0027t substitute x equals 0 because"},{"Start":"01:58.250 ","End":"02:01.790","Text":"then you\u0027ll get 0 over 0, which is undefined."},{"Start":"02:01.790 ","End":"02:04.910","Text":"Today, we have L\u0027Hopital\u0027s rule,"},{"Start":"02:04.910 ","End":"02:07.110","Text":"and most of you will have learned that,"},{"Start":"02:07.110 ","End":"02:09.440","Text":"and then it becomes trivial because we can"},{"Start":"02:09.440 ","End":"02:13.310","Text":"differentiate top and bottom and get cosine x over 1."},{"Start":"02:13.310 ","End":"02:16.650","Text":"When x goes to 0, cosine x goes to 1,"},{"Start":"02:16.650 ","End":"02:20.420","Text":"so obviously it must have been discovered before L\u0027Hopital\u0027s rule."},{"Start":"02:20.420 ","End":"02:23.000","Text":"Anyway, here it is for us to use,"},{"Start":"02:23.000 ","End":"02:26.585","Text":"and I\u0027ll show you some examples of how to use this."},{"Start":"02:26.585 ","End":"02:29.270","Text":"Because some of them like this one,"},{"Start":"02:29.270 ","End":"02:32.975","Text":"you don\u0027t even see any sine x in here anywhere."},{"Start":"02:32.975 ","End":"02:36.140","Text":"We\u0027re going to have to manipulate the exercises,"},{"Start":"02:36.140 ","End":"02:39.050","Text":"perform all kinds of algebraic tricks and"},{"Start":"02:39.050 ","End":"02:43.290","Text":"trigonometrical identities until it\u0027s brought to this form,"},{"Start":"02:43.290 ","End":"02:46.865","Text":"or at least part of it is brought to this form with some extras."},{"Start":"02:46.865 ","End":"02:50.450","Text":"You\u0027ll see in the examples what I mean."},{"Start":"02:50.450 ","End":"02:52.805","Text":"But before we get to the examples,"},{"Start":"02:52.805 ","End":"02:57.305","Text":"I\u0027d like to show you a broader viewpoint on this theorem."},{"Start":"02:57.305 ","End":"03:02.120","Text":"What I can do is look at it not just as using the letter x,"},{"Start":"03:02.120 ","End":"03:04.400","Text":"but as some general template."},{"Start":"03:04.400 ","End":"03:06.775","Text":"I\u0027ll write something down and you\u0027ll see."},{"Start":"03:06.775 ","End":"03:10.910","Text":"The limit of sine, instead of x,"},{"Start":"03:10.910 ","End":"03:13.595","Text":"I\u0027m just going to put a blank box here,"},{"Start":"03:13.595 ","End":"03:16.920","Text":"over the same blank box."},{"Start":"03:16.920 ","End":"03:25.380","Text":"This limit of sine something over the same thing is going to equal 1,"},{"Start":"03:25.380 ","End":"03:31.575","Text":"provided that this thing in the box tends to 0."},{"Start":"03:31.575 ","End":"03:36.720","Text":"It still won\u0027t look very clear until I write some examples of this,"},{"Start":"03:36.720 ","End":"03:38.645","Text":"so let\u0027s go to that next."},{"Start":"03:38.645 ","End":"03:44.330","Text":"The first illustration of this as a template is if I had limit as"},{"Start":"03:44.330 ","End":"03:50.745","Text":"x goes to 0 of sine of 4x over 4x,"},{"Start":"03:50.745 ","End":"03:52.875","Text":"that would also be 1."},{"Start":"03:52.875 ","End":"03:56.960","Text":"Because what I could do is in place of the box,"},{"Start":"03:56.960 ","End":"04:03.995","Text":"I could put 4x and I note that although the limit says x goes to 0,"},{"Start":"04:03.995 ","End":"04:09.320","Text":"when that\u0027s true, we would also have 4x goes to 0."},{"Start":"04:09.320 ","End":"04:11.735","Text":"I can highlight with a marker."},{"Start":"04:11.735 ","End":"04:14.270","Text":"Here I have, this is the important thing."},{"Start":"04:14.270 ","End":"04:22.150","Text":"This is the same expression as this and this expression also goes to 0."},{"Start":"04:22.150 ","End":"04:27.810","Text":"I\u0027d like to distinguish between this 0 and this 0."},{"Start":"04:27.810 ","End":"04:31.130","Text":"I\u0027ll do that in the following example."},{"Start":"04:31.130 ","End":"04:35.930","Text":"What if I had the limit as x goes to"},{"Start":"04:35.930 ","End":"04:42.955","Text":"1 of sine x minus 1 over x minus 1?"},{"Start":"04:42.955 ","End":"04:45.835","Text":"Notice here x doesn\u0027t go to 0."},{"Start":"04:45.835 ","End":"04:49.570","Text":"What\u0027s going to be in the box is x minus 1,"},{"Start":"04:49.570 ","End":"04:55.720","Text":"and x minus 1 does indeed go to 0 as x goes to 1,"},{"Start":"04:55.720 ","End":"04:57.070","Text":"which I won\u0027t bother writing."},{"Start":"04:57.070 ","End":"05:00.040","Text":"Therefore this limit is also equal to 1."},{"Start":"05:00.040 ","End":"05:03.640","Text":"In this case, our box was x minus 1,"},{"Start":"05:03.640 ","End":"05:08.230","Text":"which appeared here inside the sine and on the denominator,"},{"Start":"05:08.230 ","End":"05:13.665","Text":"and I also checked that it is what goes to 0, not the x."},{"Start":"05:13.665 ","End":"05:15.790","Text":"When those 3 things are there,"},{"Start":"05:15.790 ","End":"05:19.060","Text":"then I conclude that the limit is equal to 1."},{"Start":"05:19.060 ","End":"05:21.670","Text":"Maybe even a couple of more examples."},{"Start":"05:21.670 ","End":"05:25.550","Text":"Next, I\u0027ll have x goes to 2."},{"Start":"05:25.550 ","End":"05:34.295","Text":"Let\u0027s say sine of x squared minus 4 over x squared minus 4."},{"Start":"05:34.295 ","End":"05:36.640","Text":"This will also go to 1,"},{"Start":"05:36.640 ","End":"05:39.800","Text":"provided because I can see it\u0027s the same here and here,"},{"Start":"05:39.800 ","End":"05:44.900","Text":"but I still have to show that x squared minus 4 does go to 0."},{"Start":"05:44.900 ","End":"05:49.450","Text":"Well, let\u0027s check. Does x squared minus 4 go to 0?"},{"Start":"05:49.450 ","End":"05:51.140","Text":"The answer is yes,"},{"Start":"05:51.140 ","End":"05:52.775","Text":"because if x goes to 2,"},{"Start":"05:52.775 ","End":"05:54.110","Text":"I substitute 2 here,"},{"Start":"05:54.110 ","End":"05:56.645","Text":"2 squared minus 4 is 0."},{"Start":"05:56.645 ","End":"06:03.560","Text":"Once again, I have the template form where what\u0027s in the template is x squared minus 4."},{"Start":"06:03.560 ","End":"06:05.315","Text":"It\u0027s inside the sine,"},{"Start":"06:05.315 ","End":"06:07.715","Text":"it\u0027s on the denominator,"},{"Start":"06:07.715 ","End":"06:11.745","Text":"and it tends to 0."},{"Start":"06:11.745 ","End":"06:14.340","Text":"The limit as x goes to 2,"},{"Start":"06:14.340 ","End":"06:16.815","Text":"this will also equal 1."},{"Start":"06:16.815 ","End":"06:19.220","Text":"Maybe here\u0027s 1 more example of"},{"Start":"06:19.220 ","End":"06:23.015","Text":"a trigonometric limit since I have a bit of space down here."},{"Start":"06:23.015 ","End":"06:24.650","Text":"Let\u0027s try this."},{"Start":"06:24.650 ","End":"06:29.540","Text":"Limit as x goes to minus 1,"},{"Start":"06:29.540 ","End":"06:35.540","Text":"the sine of x squared minus x"},{"Start":"06:35.540 ","End":"06:43.385","Text":"minus 2 over x squared minus x minus 2."},{"Start":"06:43.385 ","End":"06:49.955","Text":"Now, let\u0027s see what happens if I let the box be x squared minus x minus 2."},{"Start":"06:49.955 ","End":"06:55.175","Text":"It\u0027s certainly the same expression in the numerator and in the denominator."},{"Start":"06:55.175 ","End":"07:00.560","Text":"The question is, what does this go to when x goes to minus 1?"},{"Start":"07:00.560 ","End":"07:03.770","Text":"Well, x squared minus x minus 2,"},{"Start":"07:03.770 ","End":"07:09.600","Text":"we can find what it goes to just by substituting minus 1."},{"Start":"07:09.600 ","End":"07:14.270","Text":"Minus 1 squared is 1 plus 1 minus 2 is 0,"},{"Start":"07:14.270 ","End":"07:17.225","Text":"so we\u0027re also all right with this,"},{"Start":"07:17.225 ","End":"07:21.335","Text":"that we can take this thing as what\u0027s in the box,"},{"Start":"07:21.335 ","End":"07:25.280","Text":"the same thing in the upstairs and downstairs,"},{"Start":"07:25.280 ","End":"07:30.105","Text":"and that\u0027s the bit that has to tend to 0."},{"Start":"07:30.105 ","End":"07:37.670","Text":"This limit will therefore also be equal to 1 as it\u0027s got all the conditions."},{"Start":"07:37.670 ","End":"07:41.195","Text":"We see that this is a bit more flexible."},{"Start":"07:41.195 ","End":"07:45.935","Text":"However, when you get a question in your homework or your exams,"},{"Start":"07:45.935 ","End":"07:50.195","Text":"you will not get it in template form just like this,"},{"Start":"07:50.195 ","End":"07:53.810","Text":"you\u0027ll be given something much more disguised like"},{"Start":"07:53.810 ","End":"07:58.535","Text":"the examples we saw in the previous page where you\u0027re going to have to do that work,"},{"Start":"07:58.535 ","End":"08:01.550","Text":"like I mentioned, using trigonometrical identities,"},{"Start":"08:01.550 ","End":"08:03.560","Text":"and a lot of algebraic manipulation,"},{"Start":"08:03.560 ","End":"08:08.490","Text":"and whatever tricks you can think of to solve these exercises."}],"ID":9617},{"Watched":false,"Name":"Trigonometric Limits Part 2","Duration":"10m 7s","ChapterTopicVideoID":9306,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.895","Text":"Now we\u0027ll get onto the real examples."},{"Start":"00:02.895 ","End":"00:05.175","Text":"Just want to clean up here a bit,"},{"Start":"00:05.175 ","End":"00:10.320","Text":"but leave this formula and we\u0027ll start with a very friendly example."},{"Start":"00:10.320 ","End":"00:19.895","Text":"We\u0027re asked what is the limit as x goes to 0 of sine 2x over x."},{"Start":"00:19.895 ","End":"00:24.050","Text":"Well already we can use this or even the template form,"},{"Start":"00:24.050 ","End":"00:29.000","Text":"not right away because we don\u0027t have the same thing top and bottom."},{"Start":"00:29.000 ","End":"00:32.345","Text":"Here it\u0027s 2x and here it\u0027s x."},{"Start":"00:32.345 ","End":"00:41.500","Text":"1 main trick that 1 uses is you first of all write it as you would ideally like it,"},{"Start":"00:41.500 ","End":"00:44.195","Text":"limit as x goes to 0."},{"Start":"00:44.195 ","End":"00:46.670","Text":"Now for it to be in the template form,"},{"Start":"00:46.670 ","End":"00:51.840","Text":"I would like to have here sine 2x and here also 2x,"},{"Start":"00:51.840 ","End":"00:55.895","Text":"and then everything will be fine because I\u0027m thinking here\u0027s 2x and here\u0027s 2x,"},{"Start":"00:55.895 ","End":"00:59.030","Text":"and when x goes to 0, this 2x also goes to 0,"},{"Start":"00:59.030 ","End":"01:00.500","Text":"so that would be ideal."},{"Start":"01:00.500 ","End":"01:03.140","Text":"The only trouble is that it\u0027s not mathematics."},{"Start":"01:03.140 ","End":"01:05.165","Text":"You can\u0027t just go about changing things,"},{"Start":"01:05.165 ","End":"01:07.010","Text":"not following the rules."},{"Start":"01:07.010 ","End":"01:10.700","Text":"I ruin it a bit at first by throwing this 2 in,"},{"Start":"01:10.700 ","End":"01:16.150","Text":"but then I make up for it and I compensate by throwing in an extra 2 at the top."},{"Start":"01:16.150 ","End":"01:18.675","Text":"If I put the extra 2 here,"},{"Start":"01:18.675 ","End":"01:20.160","Text":"the 2 cancels with the 2,"},{"Start":"01:20.160 ","End":"01:23.390","Text":"and mathematically I haven\u0027t changed the expression."},{"Start":"01:23.390 ","End":"01:26.620","Text":"However, it does look very much like the template,"},{"Start":"01:26.620 ","End":"01:30.260","Text":"because now I can say the 2 is separate."},{"Start":"01:30.260 ","End":"01:33.200","Text":"This thing is the template, like I mentioned."},{"Start":"01:33.200 ","End":"01:35.195","Text":"Here I have the 2x,"},{"Start":"01:35.195 ","End":"01:37.145","Text":"here I have the 2x,"},{"Start":"01:37.145 ","End":"01:42.900","Text":"and I note that 2x also goes to 0."},{"Start":"01:42.900 ","End":"01:44.820","Text":"This fills all the conditions,"},{"Start":"01:44.820 ","End":"01:47.430","Text":"so this part goes to 1."},{"Start":"01:47.430 ","End":"01:51.140","Text":"I can take the limit separately if this and separately if the 2,"},{"Start":"01:51.140 ","End":"01:59.965","Text":"so what I get is this is equal to 1 from the main theorem and the 2 that\u0027s leftover,"},{"Start":"01:59.965 ","End":"02:02.065","Text":"and so the answer is 2."},{"Start":"02:02.065 ","End":"02:04.909","Text":"That is what this limit equals."},{"Start":"02:04.909 ","End":"02:06.630","Text":"That\u0027s generally how we do it,"},{"Start":"02:06.630 ","End":"02:08.690","Text":"we get something that\u0027s maybe close,"},{"Start":"02:08.690 ","End":"02:10.880","Text":"maybe not so close to this form,"},{"Start":"02:10.880 ","End":"02:15.245","Text":"alter little bit, break and fix, and then substitute."},{"Start":"02:15.245 ","End":"02:19.550","Text":"We get a 1 from this limit and there\u0027ll be little bits leftover like the 2 here,"},{"Start":"02:19.550 ","End":"02:21.635","Text":"and that\u0027s the general idea."},{"Start":"02:21.635 ","End":"02:25.250","Text":"I\u0027d like to show you another way of solving this exercise."},{"Start":"02:25.250 ","End":"02:29.940","Text":"It\u0027s actually more complicated and less flexible,"},{"Start":"02:29.940 ","End":"02:33.080","Text":"but still it has some educational value."},{"Start":"02:33.080 ","End":"02:40.155","Text":"It uses trigonometrical identities and it has some value. Here goes."},{"Start":"02:40.155 ","End":"02:43.470","Text":"Someone looks at this and sees sine 2x,"},{"Start":"02:43.470 ","End":"02:47.255","Text":"and remember that this is trigonometrical identity with sine 2x,"},{"Start":"02:47.255 ","End":"02:50.495","Text":"so quite naturally, he would write,"},{"Start":"02:50.495 ","End":"02:56.420","Text":"this thing is the same as the limit as x goes to 0."},{"Start":"02:56.420 ","End":"03:05.565","Text":"Remembering the formula, this is equal to 2 sine x cosine x still over x."},{"Start":"03:05.565 ","End":"03:09.945","Text":"From here we see this sine x over x,"},{"Start":"03:09.945 ","End":"03:15.555","Text":"so we can write this as a product sine x over x times 2 cosine x."},{"Start":"03:15.555 ","End":"03:19.865","Text":"We can separate the limit of the product into separate limits,"},{"Start":"03:19.865 ","End":"03:25.640","Text":"so we have the limit as x goes to 0 for the sine x over x, but,"},{"Start":"03:25.640 ","End":"03:30.560","Text":"times the limit as x goes to 0 for the rest of it,"},{"Start":"03:30.560 ","End":"03:33.790","Text":"which is 2 cosine x."},{"Start":"03:33.790 ","End":"03:37.740","Text":"From here, this thing is equal to 1,"},{"Start":"03:37.740 ","End":"03:39.435","Text":"that\u0027s the main theorem,"},{"Start":"03:39.435 ","End":"03:41.190","Text":"and 2 cosine x."},{"Start":"03:41.190 ","End":"03:44.990","Text":"We can just substitute x equals 0 and x is 0,"},{"Start":"03:44.990 ","End":"03:47.805","Text":"that\u0027s cosine 0 is 1 times 2 is 2."},{"Start":"03:47.805 ","End":"03:49.710","Text":"Here we have 1 times 2,"},{"Start":"03:49.710 ","End":"03:52.835","Text":"and we get 2 same answer of course,"},{"Start":"03:52.835 ","End":"03:55.370","Text":"but using trigonometrical identities,"},{"Start":"03:55.370 ","End":"03:59.225","Text":"but it\u0027s not recommended, it\u0027s not generalizable."},{"Start":"03:59.225 ","End":"04:01.450","Text":"We made use of the fact that this was a 2,"},{"Start":"04:01.450 ","End":"04:02.790","Text":"if it had been, say,"},{"Start":"04:02.790 ","End":"04:04.700","Text":"sine 10x over x,"},{"Start":"04:04.700 ","End":"04:06.830","Text":"this method would have still just worked fine,"},{"Start":"04:06.830 ","End":"04:09.320","Text":"but here we couldn\u0027t have used that anymore,"},{"Start":"04:09.320 ","End":"04:11.465","Text":"still it had some value."},{"Start":"04:11.465 ","End":"04:14.890","Text":"On to the next example."},{"Start":"04:14.890 ","End":"04:17.315","Text":"Here\u0027s our next example."},{"Start":"04:17.315 ","End":"04:22.295","Text":"The limit as x goes to 0 sine 4x over sine 10x."},{"Start":"04:22.295 ","End":"04:24.440","Text":"But before we start solving this,"},{"Start":"04:24.440 ","End":"04:25.850","Text":"I wrote something here."},{"Start":"04:25.850 ","End":"04:29.420","Text":"There was a companion limit to this 1,"},{"Start":"04:29.420 ","End":"04:32.120","Text":"and it\u0027s just the inverse,"},{"Start":"04:32.120 ","End":"04:34.970","Text":"the reciprocal instead of sine x over x,"},{"Start":"04:34.970 ","End":"04:37.130","Text":"if it was x over sine x,"},{"Start":"04:37.130 ","End":"04:38.680","Text":"it would have the same limit."},{"Start":"04:38.680 ","End":"04:40.760","Text":"It\u0027s fairly easy to see why,"},{"Start":"04:40.760 ","End":"04:44.420","Text":"because this is actually 1 over this,"},{"Start":"04:44.420 ","End":"04:47.360","Text":"so the answer here should really be 1 over 1,"},{"Start":"04:47.360 ","End":"04:48.980","Text":"but that is 1."},{"Start":"04:48.980 ","End":"04:54.560","Text":"This is another limit that we sometimes use together or instead of this 1,"},{"Start":"04:54.560 ","End":"04:56.539","Text":"and we\u0027ll see it in this example."},{"Start":"04:56.539 ","End":"04:59.210","Text":"What I want to do is again,"},{"Start":"04:59.210 ","End":"05:03.770","Text":"to think of that template of sine box over box."},{"Start":"05:03.770 ","End":"05:05.330","Text":"What I would like to do,"},{"Start":"05:05.330 ","End":"05:09.260","Text":"and this is 1 way of doing these questions is, like I said,"},{"Start":"05:09.260 ","End":"05:11.200","Text":"you boil and then fix,"},{"Start":"05:11.200 ","End":"05:13.260","Text":"so limit is x goes to 0."},{"Start":"05:13.260 ","End":"05:15.050","Text":"Now sine 4x here,"},{"Start":"05:15.050 ","End":"05:18.725","Text":"what I would really like is if it was over 4x,"},{"Start":"05:18.725 ","End":"05:23.750","Text":"because then I\u0027d have exactly the template from this formula."},{"Start":"05:23.750 ","End":"05:25.730","Text":"For the next bit,"},{"Start":"05:25.730 ","End":"05:29.085","Text":"where I have sine 10x in the denominator,"},{"Start":"05:29.085 ","End":"05:31.700","Text":"what I would really like if at the top I also"},{"Start":"05:31.700 ","End":"05:34.205","Text":"had 10x because then it would look like this 1,"},{"Start":"05:34.205 ","End":"05:37.735","Text":"but with 10x replacing the x."},{"Start":"05:37.735 ","End":"05:40.040","Text":"This is all very well,"},{"Start":"05:40.040 ","End":"05:42.350","Text":"but I\u0027m changing the exercise."},{"Start":"05:42.350 ","End":"05:43.760","Text":"Now that I\u0027ve spoiled it,"},{"Start":"05:43.760 ","End":"05:45.365","Text":"now I\u0027m going to fix it again."},{"Start":"05:45.365 ","End":"05:49.350","Text":"I throw x in an extra 4x in the denominator,"},{"Start":"05:49.350 ","End":"05:51.720","Text":"so if I put a 4x in the numerator,"},{"Start":"05:51.720 ","End":"05:53.190","Text":"that will be fine,"},{"Start":"05:53.190 ","End":"05:55.310","Text":"and the 10x in the numerator,"},{"Start":"05:55.310 ","End":"06:00.065","Text":"I\u0027ll compensate by putting a 10x in the denominator."},{"Start":"06:00.065 ","End":"06:03.560","Text":"Now, notice that when x goes to 0,"},{"Start":"06:03.560 ","End":"06:07.355","Text":"4x goes to 0 and 10x goes to 0."},{"Start":"06:07.355 ","End":"06:10.490","Text":"If I can use twice the template thing,"},{"Start":"06:10.490 ","End":"06:13.040","Text":"what I mean is previously we used to say,"},{"Start":"06:13.040 ","End":"06:15.845","Text":"Okay, here\u0027s the 4x and here\u0027s the 4x,"},{"Start":"06:15.845 ","End":"06:19.700","Text":"and the 4x also goes to 0, I notice."},{"Start":"06:19.700 ","End":"06:21.950","Text":"Then in the other 1 I could say,"},{"Start":"06:21.950 ","End":"06:24.110","Text":"well, here I have the 10x,"},{"Start":"06:24.110 ","End":"06:26.330","Text":"that\u0027s like the box that we had,"},{"Start":"06:26.330 ","End":"06:31.725","Text":"and here I have 10x and the 10x goes to 0 just as well."},{"Start":"06:31.725 ","End":"06:34.865","Text":"Here in fact, we can also cancel x,"},{"Start":"06:34.865 ","End":"06:36.770","Text":"cancels with x as x isn\u0027t 0,"},{"Start":"06:36.770 ","End":"06:38.515","Text":"it\u0027s just goes to 0."},{"Start":"06:38.515 ","End":"06:42.350","Text":"If we break the limit up into product of limits,"},{"Start":"06:42.350 ","End":"06:44.630","Text":"what we\u0027ll get will be for the first bit,"},{"Start":"06:44.630 ","End":"06:48.560","Text":"will get 1 because that\u0027s exactly standard template form,"},{"Start":"06:48.560 ","End":"06:53.330","Text":"and the limit of 10x over sine 10x using this as a template,"},{"Start":"06:53.330 ","End":"06:54.935","Text":"we\u0027ll also be cool 1,"},{"Start":"06:54.935 ","End":"06:56.945","Text":"and this is just 4/10."},{"Start":"06:56.945 ","End":"07:01.140","Text":"You can write it as a decimal of 0.4 or 2/5,"},{"Start":"07:01.140 ","End":"07:04.355","Text":"in any event, I\u0027ll just leave it as 4/10."},{"Start":"07:04.355 ","End":"07:07.480","Text":"Let\u0027s go on to the next exercise."},{"Start":"07:07.480 ","End":"07:09.630","Text":"Here\u0027s our next exercise."},{"Start":"07:09.630 ","End":"07:12.300","Text":"Limit x goes to 0,"},{"Start":"07:12.300 ","End":"07:15.440","Text":"1 minus cosine 2x over x squared."},{"Start":"07:15.440 ","End":"07:19.700","Text":"I don\u0027t know how well each 1 remembers trigonometrical identities,"},{"Start":"07:19.700 ","End":"07:21.900","Text":"I\u0027ll take it that you know some of them,"},{"Start":"07:21.900 ","End":"07:26.735","Text":"and I recommend looking them up if they\u0027re very useful for these trigonometric limits."},{"Start":"07:26.735 ","End":"07:31.280","Text":"Some people even know the final formula for 1 minus cosine 2x."},{"Start":"07:31.280 ","End":"07:35.660","Text":"Now assume that you\u0027ve learned this in 1 of several forms,"},{"Start":"07:35.660 ","End":"07:40.765","Text":"I actually tend to remember all 3 forms of cosine 2x."},{"Start":"07:40.765 ","End":"07:42.795","Text":"When I learned cosine 2x,"},{"Start":"07:42.795 ","End":"07:44.660","Text":"we learned in original form,"},{"Start":"07:44.660 ","End":"07:51.705","Text":"which was cosine squared x minus sine squared x,"},{"Start":"07:51.705 ","End":"07:54.184","Text":"but there were 2 other forms."},{"Start":"07:54.184 ","End":"07:59.775","Text":"There was another form which was 2 cosine squared x minus 1,"},{"Start":"07:59.775 ","End":"08:01.320","Text":"just use cosines,"},{"Start":"08:01.320 ","End":"08:03.190","Text":"and if you just want sines,"},{"Start":"08:03.190 ","End":"08:07.490","Text":"then it\u0027s 1 minus 2 sine squared x."},{"Start":"08:07.490 ","End":"08:09.110","Text":"This is the 1 that I want,"},{"Start":"08:09.110 ","End":"08:11.255","Text":"the sine x, of course."},{"Start":"08:11.255 ","End":"08:18.590","Text":"I will write it in the form that this equals the limit as x goes to 0,"},{"Start":"08:18.590 ","End":"08:22.640","Text":"1 minus, and instead of the cosine 2x,"},{"Start":"08:22.640 ","End":"08:24.020","Text":"then as I say,"},{"Start":"08:24.020 ","End":"08:28.655","Text":"I\u0027ll be using this form over x squared."},{"Start":"08:28.655 ","End":"08:30.695","Text":"Now, just a little bit of algebra,"},{"Start":"08:30.695 ","End":"08:35.420","Text":"this is equal to limit as x goes to 0."},{"Start":"08:35.420 ","End":"08:42.290","Text":"1 minus just makes it to sine squared x over x squared,"},{"Start":"08:42.290 ","End":"08:45.685","Text":"this equals the limit of 2."},{"Start":"08:45.685 ","End":"08:49.205","Text":"Now here I can see my sine x over x is just what I wanted."},{"Start":"08:49.205 ","End":"08:52.695","Text":"Sine x over x all this squared,"},{"Start":"08:52.695 ","End":"08:54.495","Text":"and that\u0027s the same thing,"},{"Start":"08:54.495 ","End":"08:58.835","Text":"and then if I just take each piece separately,"},{"Start":"08:58.835 ","End":"09:02.240","Text":"I can put the limit basically inside the sine x over x."},{"Start":"09:02.240 ","End":"09:06.830","Text":"Another way you could do it is write it as sine x over x times another sine x over x."},{"Start":"09:06.830 ","End":"09:11.345","Text":"But it\u0027s easy, just take the limit of sine x over x, which is 1,"},{"Start":"09:11.345 ","End":"09:16.325","Text":"and this will equal 2 times 1 squared,"},{"Start":"09:16.325 ","End":"09:19.210","Text":"and so the answer is 2."},{"Start":"09:19.210 ","End":"09:21.915","Text":"Before I finish the lesson,"},{"Start":"09:21.915 ","End":"09:27.250","Text":"I just like to show you that besides these 2 basic templates for limits,"},{"Start":"09:27.250 ","End":"09:30.710","Text":"there\u0027s a couple of others that are very similar to this,"},{"Start":"09:30.710 ","End":"09:32.390","Text":"except that instead of the sine,"},{"Start":"09:32.390 ","End":"09:37.490","Text":"I have the tangent and they look the same except replace sine by tangent,"},{"Start":"09:37.490 ","End":"09:39.725","Text":"and these are also limits that work."},{"Start":"09:39.725 ","End":"09:41.945","Text":"It\u0027s not hard to see why."},{"Start":"09:41.945 ","End":"09:44.270","Text":"You don\u0027t have to know why, but basically,"},{"Start":"09:44.270 ","End":"09:48.080","Text":"the ratio between the sine and the tangent is a cosine."},{"Start":"09:48.080 ","End":"09:50.315","Text":"Because tangent to sine over cosine,"},{"Start":"09:50.315 ","End":"09:54.165","Text":"and because cosine goes to 1 when x goes to 0."},{"Start":"09:54.165 ","End":"09:55.760","Text":"These 2 limits work also,"},{"Start":"09:55.760 ","End":"09:58.505","Text":"but you can just take them without proof."},{"Start":"09:58.505 ","End":"10:00.480","Text":"These 2 will also work."},{"Start":"10:00.480 ","End":"10:03.200","Text":"These are the 4 basic limits that you\u0027ll be using to"},{"Start":"10:03.200 ","End":"10:07.750","Text":"solve trigonometric limits. That\u0027s it."}],"ID":9618},{"Watched":false,"Name":"Exercise 1","Duration":"2m 50s","ChapterTopicVideoID":4782,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.860","Text":"In this exercise, we have to find the limit as x tends to 0 of sine of 3x over 4x."},{"Start":"00:07.860 ","End":"00:11.295","Text":"Now, although this is an elementary function,"},{"Start":"00:11.295 ","End":"00:14.400","Text":"we can\u0027t just substitute x equals 0,"},{"Start":"00:14.400 ","End":"00:15.735","Text":"would have been nice,"},{"Start":"00:15.735 ","End":"00:20.730","Text":"but it doesn\u0027t work because the denominator 4x becomes 0"},{"Start":"00:20.730 ","End":"00:26.220","Text":"when x is 0 and 3x also would be 0 and sine of 0 is 0."},{"Start":"00:26.220 ","End":"00:31.860","Text":"In other words, a direct substitution would lead us to 0 over 0,"},{"Start":"00:31.860 ","End":"00:33.195","Text":"which doesn\u0027t make sense,"},{"Start":"00:33.195 ","End":"00:35.535","Text":"so we\u0027re going to have to use another technique."},{"Start":"00:35.535 ","End":"00:37.905","Text":"Here we see the sine function,"},{"Start":"00:37.905 ","End":"00:42.230","Text":"and 1 of the most basic limits in trigonometry"},{"Start":"00:42.230 ","End":"00:46.685","Text":"is that the limit of the sine of something,"},{"Start":"00:46.685 ","End":"00:53.340","Text":"let\u0027s just call it smiley over smiley."},{"Start":"00:53.340 ","End":"00:59.510","Text":"When smiley goes to 0 is equal to 1."},{"Start":"00:59.510 ","End":"01:01.640","Text":"Otherwise, if something goes tends to 0,"},{"Start":"01:01.640 ","End":"01:05.000","Text":"the sine of that something go over that something, goes to 1."},{"Start":"01:05.000 ","End":"01:11.030","Text":"Now in our case, it\u0027s natural for the smiley to be 3x because when x goes to 0,"},{"Start":"01:11.030 ","End":"01:13.180","Text":"3x goes to 0."},{"Start":"01:13.180 ","End":"01:20.390","Text":"In that case, what we would like to have is the limit as x goes to 0,"},{"Start":"01:20.390 ","End":"01:29.890","Text":"which is the same as 3x going to 0 of the sine of 3x over 3x."},{"Start":"01:29.890 ","End":"01:31.205","Text":"If this is what we had,"},{"Start":"01:31.205 ","End":"01:32.270","Text":"that would be just great,"},{"Start":"01:32.270 ","End":"01:33.950","Text":"the answer would be 1."},{"Start":"01:33.950 ","End":"01:35.650","Text":"But it\u0027s not quite."},{"Start":"01:35.650 ","End":"01:38.415","Text":"We have 3x here and here we don\u0027t."},{"Start":"01:38.415 ","End":"01:41.480","Text":"What we have to do, we have to compensate."},{"Start":"01:41.480 ","End":"01:44.495","Text":"We divide it by 3x when we shouldn\u0027t have,"},{"Start":"01:44.495 ","End":"01:50.840","Text":"so we need to put 3x in the numerator and then it\u0027s as if this cancels out."},{"Start":"01:50.840 ","End":"01:53.300","Text":"Also, we had here a 4x,"},{"Start":"01:53.300 ","End":"01:55.590","Text":"we can\u0027t take that away."},{"Start":"01:55.590 ","End":"01:58.540","Text":"In other words, this is the same as this."},{"Start":"01:58.540 ","End":"02:01.405","Text":"Once we cancel out the 3x over 3x,"},{"Start":"02:01.405 ","End":"02:03.250","Text":"which is equal to 1."},{"Start":"02:03.250 ","End":"02:09.005","Text":"Now, 3x over 4x is also can be canceled, that\u0027s just 3/4."},{"Start":"02:09.005 ","End":"02:12.345","Text":"Remember, x is not 0, it only tends to 0."},{"Start":"02:12.345 ","End":"02:15.145","Text":"When you have a constant times a limit,"},{"Start":"02:15.145 ","End":"02:17.605","Text":"the constant comes outside the limit."},{"Start":"02:17.605 ","End":"02:24.265","Text":"This is equal to 3/4 of the limit as"},{"Start":"02:24.265 ","End":"02:31.245","Text":"x goes to 0 of sine 3x over 3x."},{"Start":"02:31.245 ","End":"02:33.490","Text":"Like we said before,"},{"Start":"02:33.490 ","End":"02:38.215","Text":"this applies in this category of sines smiley over smiley,"},{"Start":"02:38.215 ","End":"02:42.045","Text":"so this whole thing becomes 1, the limit."},{"Start":"02:42.045 ","End":"02:44.550","Text":"Then we have 3/4 of the limit,"},{"Start":"02:44.550 ","End":"02:46.815","Text":"3/4 times 1,"},{"Start":"02:46.815 ","End":"02:51.190","Text":"so the answer is just 3/4 and that\u0027s it."}],"ID":4790},{"Watched":false,"Name":"Exercise 2","Duration":"3m 2s","ChapterTopicVideoID":4783,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.260","Text":"In this exercise, we have to find the limit as x goes to 0 of sine 3x over sine 4x."},{"Start":"00:07.260 ","End":"00:10.410","Text":"We\u0027d like to substitute x equals 0,"},{"Start":"00:10.410 ","End":"00:13.005","Text":"but here we can\u0027t because if x is 0,"},{"Start":"00:13.005 ","End":"00:14.730","Text":"4x and 3x is 0,"},{"Start":"00:14.730 ","End":"00:16.530","Text":"and the sine of 0 is 0."},{"Start":"00:16.530 ","End":"00:20.760","Text":"In other words, this would be of the form 0 over 0,"},{"Start":"00:20.760 ","End":"00:23.540","Text":"which is an indeterminate form, doesn\u0027t make sense."},{"Start":"00:23.540 ","End":"00:25.600","Text":"We\u0027ll then have to use another technique."},{"Start":"00:25.600 ","End":"00:28.950","Text":"Now we see that the sine function is mentioned."},{"Start":"00:28.950 ","End":"00:30.430","Text":"We think a trigonometry,"},{"Start":"00:30.430 ","End":"00:33.155","Text":"and the most basic limit in trigonometry,"},{"Start":"00:33.155 ","End":"00:34.865","Text":"and I\u0027ll write it at the side,"},{"Start":"00:34.865 ","End":"00:38.565","Text":"that if you have the sine of something,"},{"Start":"00:38.565 ","End":"00:45.905","Text":"a sine of smiley over smiley has a limit when the smiley goes to 0,"},{"Start":"00:45.905 ","End":"00:48.140","Text":"the limit is 1."},{"Start":"00:48.140 ","End":"00:52.460","Text":"That\u0027s one of the most basic trigonometric formulae for limits."},{"Start":"00:52.460 ","End":"00:58.915","Text":"It also happens to be the case that if instead of this I wrote fraction in upside down,"},{"Start":"00:58.915 ","End":"01:04.425","Text":"but smiley over sine of smiley are the same thing."},{"Start":"01:04.425 ","End":"01:06.360","Text":"The limit would also be 1."},{"Start":"01:06.360 ","End":"01:11.620","Text":"The same limit where the smiley goes to 0."},{"Start":"01:11.620 ","End":"01:13.080","Text":"That\u0027s a reminder."},{"Start":"01:13.080 ","End":"01:15.455","Text":"Now let\u0027s see how this works in our case."},{"Start":"01:15.455 ","End":"01:17.390","Text":"In our case, on the one hand,"},{"Start":"01:17.390 ","End":"01:19.160","Text":"we want the smiley to be 3x,"},{"Start":"01:19.160 ","End":"01:21.915","Text":"on the other hand, 4x. Let\u0027s see."},{"Start":"01:21.915 ","End":"01:24.660","Text":"We have the limit x goes to 0."},{"Start":"01:24.660 ","End":"01:26.510","Text":"But do note that when x goes to 0,"},{"Start":"01:26.510 ","End":"01:29.135","Text":"3x goes to 0 so does 4x."},{"Start":"01:29.135 ","End":"01:31.235","Text":"This we can leave as x."},{"Start":"01:31.235 ","End":"01:33.170","Text":"Now, what I\u0027d like to have,"},{"Start":"01:33.170 ","End":"01:37.745","Text":"if I had sine of 3x over 3x,"},{"Start":"01:37.745 ","End":"01:39.245","Text":"that would be great."},{"Start":"01:39.245 ","End":"01:45.115","Text":"Also, if I had 4x over sine 4x,"},{"Start":"01:45.115 ","End":"01:46.985","Text":"that would also be great."},{"Start":"01:46.985 ","End":"01:50.240","Text":"But this is not the same exercises as we\u0027ve changed it."},{"Start":"01:50.240 ","End":"01:51.785","Text":"Let\u0027s see what we\u0027ve done."},{"Start":"01:51.785 ","End":"01:54.545","Text":"Sine 3x over sine 4x we have from here."},{"Start":"01:54.545 ","End":"01:57.095","Text":"We\u0027ve multiplied the numerator by 4x."},{"Start":"01:57.095 ","End":"01:59.105","Text":"We\u0027ve divided by 3x."},{"Start":"01:59.105 ","End":"02:00.895","Text":"We have to compensate."},{"Start":"02:00.895 ","End":"02:05.700","Text":"The denominator of 3x has to go with the numerator of 3x."},{"Start":"02:05.700 ","End":"02:10.340","Text":"The numerator of 4x we have to compensate by dividing by 4x."},{"Start":"02:10.340 ","End":"02:14.265","Text":"In essence, we have 2 cancellations."},{"Start":"02:14.265 ","End":"02:16.325","Text":"This cancels with this,"},{"Start":"02:16.325 ","End":"02:17.960","Text":"and that cancels with that,"},{"Start":"02:17.960 ","End":"02:19.865","Text":"and that\u0027s the original exercise."},{"Start":"02:19.865 ","End":"02:23.930","Text":"Now, this helps us because when we have a limit of a product,"},{"Start":"02:23.930 ","End":"02:26.450","Text":"we can take each piece separately."},{"Start":"02:26.450 ","End":"02:31.820","Text":"Before that I just also like to mention that we can cancel the x here."},{"Start":"02:31.820 ","End":"02:34.850","Text":"Natural thing to do because x is not equal to 0,"},{"Start":"02:34.850 ","End":"02:36.440","Text":"it only tends to 0."},{"Start":"02:36.440 ","End":"02:38.675","Text":"Now we have 3 pieces."},{"Start":"02:38.675 ","End":"02:41.200","Text":"This piece, this piece, and this piece."},{"Start":"02:41.200 ","End":"02:43.055","Text":"We take the limit for each one."},{"Start":"02:43.055 ","End":"02:46.370","Text":"This part over this is sine smiley over smiley."},{"Start":"02:46.370 ","End":"02:47.450","Text":"That\u0027s 1."},{"Start":"02:47.450 ","End":"02:50.410","Text":"This thing is smiley over sine smiley."},{"Start":"02:50.410 ","End":"02:52.770","Text":"This is also 1."},{"Start":"02:52.770 ","End":"02:55.110","Text":"This is 3 quarters."},{"Start":"02:55.110 ","End":"02:58.215","Text":"Altogether, 1 times 1 times 3 quarters."},{"Start":"02:58.215 ","End":"03:03.280","Text":"The answer is 3 quarters. We\u0027re done."}],"ID":4791},{"Watched":false,"Name":"Exercise 3","Duration":"2m 43s","ChapterTopicVideoID":4785,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this exercise, we have to find the limit as x"},{"Start":"00:04.200 ","End":"00:08.985","Text":"tends to 0 of x cosine of x over sine 2x."},{"Start":"00:08.985 ","End":"00:10.710","Text":"It\u0027s an elementary function."},{"Start":"00:10.710 ","End":"00:14.970","Text":"The first thing we try is just to substitute and see if that works."},{"Start":"00:14.970 ","End":"00:18.585","Text":"Well, if we put x equals 0 here,"},{"Start":"00:18.585 ","End":"00:21.495","Text":"then the numerator is 0."},{"Start":"00:21.495 ","End":"00:24.390","Text":"If we put x equal to 0 here,"},{"Start":"00:24.390 ","End":"00:26.895","Text":"we get sine of 0, which is also 0."},{"Start":"00:26.895 ","End":"00:29.775","Text":"So we get the 0 over 0 form,"},{"Start":"00:29.775 ","End":"00:33.090","Text":"which means that the simple substitution doesn\u0027t work."},{"Start":"00:33.090 ","End":"00:35.340","Text":"Going to have to use some other techniques."},{"Start":"00:35.340 ","End":"00:37.770","Text":"Let\u0027s first copy the exercise,"},{"Start":"00:37.770 ","End":"00:39.510","Text":"we\u0027re going to need some formula,"},{"Start":"00:39.510 ","End":"00:44.810","Text":"and one of the basic trigonometric limits is the following."},{"Start":"00:44.810 ","End":"00:46.910","Text":"The limit of something,"},{"Start":"00:46.910 ","End":"00:48.035","Text":"call it smiley,"},{"Start":"00:48.035 ","End":"00:53.735","Text":"limit of smiley goes to 0 of smiley over sine smiley equals 1."},{"Start":"00:53.735 ","End":"01:00.185","Text":"What I would like to do is to use this here where my smiley is 2x."},{"Start":"01:00.185 ","End":"01:04.655","Text":"So let\u0027s rewrite this in a different form."},{"Start":"01:04.655 ","End":"01:09.860","Text":"Limit as x goes to 0."},{"Start":"01:09.860 ","End":"01:11.690","Text":"Now in the denominator,"},{"Start":"01:11.690 ","End":"01:16.220","Text":"I have sine 2x and if 2x is the smiley here,"},{"Start":"01:16.220 ","End":"01:18.725","Text":"what I\u0027d like to put here is 2x."},{"Start":"01:18.725 ","End":"01:21.200","Text":"But I\u0027ve changed the exercise from this to this,"},{"Start":"01:21.200 ","End":"01:22.220","Text":"I have to compensate."},{"Start":"01:22.220 ","End":"01:25.430","Text":"Well, first thing I\u0027ve done is forgotten the cosine of x."},{"Start":"01:25.430 ","End":"01:26.880","Text":"So let\u0027s put it back in,"},{"Start":"01:26.880 ","End":"01:31.090","Text":"and the other thing is that here I have x and here I have 2x."},{"Start":"01:31.090 ","End":"01:37.500","Text":"So I have to put a 2 here so that the 2 and the 2 cancel."},{"Start":"01:37.500 ","End":"01:43.145","Text":"Now I\u0027m going to be able to use this formula with my smiley as 2x."},{"Start":"01:43.145 ","End":"01:46.685","Text":"Let\u0027s write it in a slightly more convenient form."},{"Start":"01:46.685 ","End":"01:53.435","Text":"Limit x tends to 0 and the 2x over the sine 2x,"},{"Start":"01:53.435 ","End":"01:57.885","Text":"I\u0027ll write separately 2x over"},{"Start":"01:57.885 ","End":"02:05.185","Text":"sine 2x times what remains is cosine x over 2."},{"Start":"02:05.185 ","End":"02:06.920","Text":"Now, for the first part,"},{"Start":"02:06.920 ","End":"02:08.585","Text":"I\u0027m going to use this formula."},{"Start":"02:08.585 ","End":"02:11.990","Text":"I should have mentioned that smiley tends to 0."},{"Start":"02:11.990 ","End":"02:14.360","Text":"It should really be 2x tends to 0,"},{"Start":"02:14.360 ","End":"02:17.670","Text":"but that\u0027s the same thing as x tends to 0."},{"Start":"02:17.740 ","End":"02:21.260","Text":"So this is equal 2,"},{"Start":"02:21.260 ","End":"02:26.405","Text":"we take the limit of the first part and that\u0027s equal to 1 times."},{"Start":"02:26.405 ","End":"02:28.790","Text":"Here we can just substitute x equals 0."},{"Start":"02:28.790 ","End":"02:32.720","Text":"Cosine x is 1 and the 2 in the denominator."},{"Start":"02:32.720 ","End":"02:35.645","Text":"So we have 1 times a 1/2,"},{"Start":"02:35.645 ","End":"02:38.970","Text":"and the answer is 1.5,"},{"Start":"02:38.970 ","End":"02:41.045","Text":"simply that."},{"Start":"02:41.045 ","End":"02:43.530","Text":"That\u0027s our answer."}],"ID":4792},{"Watched":false,"Name":"Exercise 3 - Alternate","Duration":"1m 34s","ChapterTopicVideoID":4784,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.855","Text":"I\u0027d like to mention that there is another way to solve this exercise,"},{"Start":"00:03.855 ","End":"00:07.575","Text":"using another trigonometrical formula for sine 2x."},{"Start":"00:07.575 ","End":"00:11.460","Text":"This time I\u0027m going to use the formula for sine 2x."},{"Start":"00:11.460 ","End":"00:13.470","Text":"Let me just write it at the side."},{"Start":"00:13.470 ","End":"00:15.825","Text":"This is the formula."},{"Start":"00:15.825 ","End":"00:18.135","Text":"If I use it here,"},{"Start":"00:18.135 ","End":"00:25.575","Text":"I will get limit x tends to 0 of"},{"Start":"00:25.575 ","End":"00:30.945","Text":"x cosine x over"},{"Start":"00:30.945 ","End":"00:38.450","Text":"copying from here 2 sine x, cosine x."},{"Start":"00:38.450 ","End":"00:48.740","Text":"The cosine x cancels and what I\u0027m left with is the limit as x goes to 0."},{"Start":"00:48.740 ","End":"00:57.950","Text":"Write it slightly differently as x over sine x times 1.5,"},{"Start":"00:57.950 ","End":"00:59.480","Text":"I\u0027ll take the 2 aside."},{"Start":"00:59.480 ","End":"01:03.850","Text":"I did this because this thing here looks exactly like this,"},{"Start":"01:03.850 ","End":"01:07.520","Text":"this time with a smiley being sine x."},{"Start":"01:07.520 ","End":"01:12.210","Text":"This limit becomes, this thing is 1."},{"Start":"01:12.210 ","End":"01:18.425","Text":"This thing stays 1.5 and our final answer is 1.5,"},{"Start":"01:18.425 ","End":"01:20.525","Text":"which is the same as before."},{"Start":"01:20.525 ","End":"01:24.440","Text":"But I commend the previous method because for example,"},{"Start":"01:24.440 ","End":"01:25.490","Text":"if this wasn\u0027t a 2,"},{"Start":"01:25.490 ","End":"01:27.470","Text":"it was 3 or 4 or any other number,"},{"Start":"01:27.470 ","End":"01:30.560","Text":"it wouldn\u0027t have such a convenient formula as this,"},{"Start":"01:30.560 ","End":"01:35.370","Text":"so I recommend the previous method. That\u0027s all."}],"ID":4793},{"Watched":false,"Name":"Exercise 4","Duration":"3m 44s","ChapterTopicVideoID":4786,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"In this exercise, we have to find the limit as x tends to 0 of 1"},{"Start":"00:05.220 ","End":"00:10.770","Text":"minus cosine x over x squared, an elementary function."},{"Start":"00:10.770 ","End":"00:14.655","Text":"The first thing we try is to just substitute x equals 0."},{"Start":"00:14.655 ","End":"00:16.950","Text":"If that works, then fine."},{"Start":"00:16.950 ","End":"00:22.065","Text":"Unfortunately, it doesn\u0027t because if we put x is 0,"},{"Start":"00:22.065 ","End":"00:23.895","Text":"0 squared is 0."},{"Start":"00:23.895 ","End":"00:26.010","Text":"Also, the cosine of 0 is 1,"},{"Start":"00:26.010 ","End":"00:27.750","Text":"1 minus 1 is 0."},{"Start":"00:27.750 ","End":"00:32.430","Text":"This is 1 of those limits of the form 0 over 0,"},{"Start":"00:32.430 ","End":"00:35.015","Text":"so we have to use some other technique,"},{"Start":"00:35.015 ","End":"00:37.655","Text":"trigonometric formula, and so on."},{"Start":"00:37.655 ","End":"00:39.800","Text":"The 1 that comes to mind,"},{"Start":"00:39.800 ","End":"00:43.220","Text":"the most famous or useful trigonometrical formula,"},{"Start":"00:43.220 ","End":"00:44.345","Text":"I\u0027ll write it here,"},{"Start":"00:44.345 ","End":"00:48.570","Text":"is that the limit as something,"},{"Start":"00:48.570 ","End":"00:50.335","Text":"let\u0027s call it smiley,"},{"Start":"00:50.335 ","End":"01:00.170","Text":"tends to 0 of sine smiley over smiley is equal to 1."},{"Start":"01:00.170 ","End":"01:05.655","Text":"Unfortunately, we don\u0027t see any sine, only cosines."},{"Start":"01:05.655 ","End":"01:09.150","Text":"Nothing of the sort from here and here."},{"Start":"01:09.150 ","End":"01:11.210","Text":"What we\u0027re going to do is use another"},{"Start":"01:11.210 ","End":"01:14.705","Text":"trigonometric formula for this numerator,"},{"Start":"01:14.705 ","End":"01:16.475","Text":"and I\u0027ll write it down at the side,"},{"Start":"01:16.475 ","End":"01:18.035","Text":"and this is the formula."},{"Start":"01:18.035 ","End":"01:22.040","Text":"What I\u0027m going to do is now copy the original exercise first."},{"Start":"01:22.040 ","End":"01:25.945","Text":"Now I\u0027m going to use this second formula here,"},{"Start":"01:25.945 ","End":"01:32.555","Text":"and this is equal to the limit x goes to 0 of"},{"Start":"01:32.555 ","End":"01:42.890","Text":"2 sine squared x over 2 over x squared."},{"Start":"01:42.890 ","End":"01:45.200","Text":"Just to make it more convenient"},{"Start":"01:45.200 ","End":"01:49.150","Text":"and the square might be confusing,"},{"Start":"01:49.150 ","End":"01:53.435","Text":"so I\u0027ll just write the square as something times itself."},{"Start":"01:53.435 ","End":"02:00.665","Text":"It\u0027s going to equal 2 times sine x over 2."},{"Start":"02:00.665 ","End":"02:07.860","Text":"Again, times sine x over 2 over,"},{"Start":"02:07.860 ","End":"02:12.709","Text":"and this x squared I\u0027ll write as x times x."},{"Start":"02:12.709 ","End":"02:15.695","Text":"Now this is beginning to look more like this."},{"Start":"02:15.695 ","End":"02:17.330","Text":"In this formula, I have to have the"},{"Start":"02:17.330 ","End":"02:19.865","Text":"same thing here and here."},{"Start":"02:19.865 ","End":"02:21.605","Text":"But here I don\u0027t."},{"Start":"02:21.605 ","End":"02:24.290","Text":"This is x over 2 and this is x."},{"Start":"02:24.290 ","End":"02:27.320","Text":"We\u0027ll first spoil it and then we\u0027ll fix it."},{"Start":"02:27.320 ","End":"02:32.980","Text":"This is equal to limit x goes to 0 of"},{"Start":"02:32.980 ","End":"02:40.510","Text":"2 sine x over 2 over,"},{"Start":"02:40.510 ","End":"02:46.140","Text":"what I\u0027m going to do is change this x to x over 2."},{"Start":"02:46.140 ","End":"02:50.560","Text":"Also, I\u0027ll change this x to x over 2."},{"Start":"02:50.560 ","End":"02:54.185","Text":"But I can\u0027t just do that because I\u0027ve changed the exercise."},{"Start":"02:54.185 ","End":"02:57.655","Text":"I\u0027ve divided by 2 in the denominator and again by 2,"},{"Start":"02:57.655 ","End":"03:00.015","Text":"so what I have to do is to compensate."},{"Start":"03:00.015 ","End":"03:03.740","Text":"What I\u0027ll do is compensate by writing a 4 here"},{"Start":"03:03.740 ","End":"03:05.915","Text":"and then that fixes the 2 and the 2."},{"Start":"03:05.915 ","End":"03:09.740","Text":"I\u0027ve highlighted the smiley part."},{"Start":"03:09.740 ","End":"03:13.120","Text":"What I have is 2 over 4."},{"Start":"03:13.120 ","End":"03:16.845","Text":"Here I have what looks like this,"},{"Start":"03:16.845 ","End":"03:20.435","Text":"and here I have another 1 that looks like that."},{"Start":"03:20.435 ","End":"03:22.460","Text":"When I take the limit,"},{"Start":"03:22.460 ","End":"03:29.010","Text":"what I get is 2 over 4, which is 1/2."},{"Start":"03:29.010 ","End":"03:35.980","Text":"The sine x over 2 over x over 2 goes to 1 by this formula."},{"Start":"03:36.470 ","End":"03:39.195","Text":"The same thing is also 1,"},{"Start":"03:39.195 ","End":"03:45.210","Text":"so the final answer is 1/2. This is it."}],"ID":4794},{"Watched":false,"Name":"Exercise 5","Duration":"2m 59s","ChapterTopicVideoID":4787,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this exercise, we have to find the limit as x tends to"},{"Start":"00:03.690 ","End":"00:07.845","Text":"0 of tangent x minus sine x over x cubed."},{"Start":"00:07.845 ","End":"00:10.770","Text":"First, we try substituting x equals 0."},{"Start":"00:10.770 ","End":"00:12.300","Text":"This is an elementary function,"},{"Start":"00:12.300 ","End":"00:13.890","Text":"so that\u0027s what we would normally do."},{"Start":"00:13.890 ","End":"00:18.665","Text":"Substituting gives us 0 minus 0 over 0,"},{"Start":"00:18.665 ","End":"00:21.750","Text":"so we get something of the form 0 over 0,"},{"Start":"00:21.750 ","End":"00:24.195","Text":"so substitution is not going to work."},{"Start":"00:24.195 ","End":"00:26.565","Text":"We\u0027re going to have to use some techniques."},{"Start":"00:26.565 ","End":"00:28.650","Text":"The obvious thing to do is use 1"},{"Start":"00:28.650 ","End":"00:31.755","Text":"of the trigonometric limit formulas."},{"Start":"00:31.755 ","End":"00:34.260","Text":"Do this as a side exercise."},{"Start":"00:34.260 ","End":"00:40.040","Text":"Tangent x minus sine x is equal to,"},{"Start":"00:40.040 ","End":"00:42.195","Text":"tangent is sine over cosine,"},{"Start":"00:42.195 ","End":"00:52.445","Text":"so we have sine x over cosine x minus sine x."},{"Start":"00:52.445 ","End":"00:55.670","Text":"Now if we do a bit of algebra and write it"},{"Start":"00:55.670 ","End":"01:00.540","Text":"over a common denominator cosine x,"},{"Start":"01:00.540 ","End":"01:03.180","Text":"from here we get sine x,"},{"Start":"01:03.180 ","End":"01:04.970","Text":"and from here, because we multiply"},{"Start":"01:04.970 ","End":"01:06.640","Text":"the denominator by cosine x,"},{"Start":"01:06.640 ","End":"01:13.730","Text":"we\u0027ll have to write it as cosine x times sine x."},{"Start":"01:13.730 ","End":"01:19.400","Text":"Here, we can take sine x outside the brackets and write it as"},{"Start":"01:19.400 ","End":"01:29.730","Text":"sine x times 1 minus cosine x all over cosine x."},{"Start":"01:29.730 ","End":"01:33.050","Text":"Now, I\u0027m going to put this expression back in here"},{"Start":"01:33.050 ","End":"01:35.030","Text":"instead of what I highlighted,"},{"Start":"01:35.030 ","End":"01:39.290","Text":"and we\u0027ll get the limit x tends to"},{"Start":"01:39.290 ","End":"01:46.585","Text":"0 of sine x brackets 1 minus cosine x."},{"Start":"01:46.585 ","End":"01:48.830","Text":"Now, the denominator here, I can join"},{"Start":"01:48.830 ","End":"01:51.755","Text":"with the denominator here,"},{"Start":"01:51.755 ","End":"01:58.350","Text":"so there\u0027s the cosine x from here times x cubed,"},{"Start":"01:58.350 ","End":"02:02.720","Text":"but I\u0027d rather write the x cubed as x times x squared."},{"Start":"02:02.720 ","End":"02:06.430","Text":"The reason for that is I can see the sine x over x."},{"Start":"02:06.430 ","End":"02:11.390","Text":"Now, we can divide this as follows, into 3 pieces."},{"Start":"02:11.390 ","End":"02:13.655","Text":"First, sine x over x,"},{"Start":"02:13.655 ","End":"02:17.445","Text":"because we know from this formula what the limit of that is."},{"Start":"02:17.445 ","End":"02:20.805","Text":"Next, that takes care of this, and this."},{"Start":"02:20.805 ","End":"02:24.440","Text":"Next, 1 over cosine x,"},{"Start":"02:24.440 ","End":"02:26.120","Text":"to take care of that,"},{"Start":"02:26.120 ","End":"02:30.690","Text":"1 over cosine x. Lastly,"},{"Start":"02:30.690 ","End":"02:33.810","Text":"1 minus cosine x over x squared,"},{"Start":"02:33.810 ","End":"02:39.195","Text":"this is equal to sine x over x goes to 1."},{"Start":"02:39.195 ","End":"02:41.180","Text":"1 over cosine x,"},{"Start":"02:41.180 ","End":"02:44.555","Text":"when we substitute x equals 0 is 1."},{"Start":"02:44.555 ","End":"02:48.960","Text":"The final bit appeared in the previous exercise,"},{"Start":"02:48.960 ","End":"02:52.095","Text":"and we found that this was equal to 1/2."},{"Start":"02:52.095 ","End":"02:54.215","Text":"Altogether, multiplying these,"},{"Start":"02:54.215 ","End":"02:59.460","Text":"the final answer is 1/2, and we\u0027re done."}],"ID":4795},{"Watched":false,"Name":"Exercise 6","Duration":"7m 44s","ChapterTopicVideoID":29845,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":31562},{"Watched":false,"Name":"Exercise 7","Duration":"6m 58s","ChapterTopicVideoID":4789,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.225","Text":"In this exercise, we have to find the limit as x tends to 0 of this expression."},{"Start":"00:06.225 ","End":"00:08.280","Text":"This is an elementary function,"},{"Start":"00:08.280 ","End":"00:12.465","Text":"so the first thing to do is to try and substitute x equals 0."},{"Start":"00:12.465 ","End":"00:14.730","Text":"If we do that, if x is 0,"},{"Start":"00:14.730 ","End":"00:18.120","Text":"x to the 4th is also equal to 0."},{"Start":"00:18.120 ","End":"00:20.400","Text":"Since cosine of x is 1,"},{"Start":"00:20.400 ","End":"00:22.530","Text":"here we have 1 minus 1 is 0,"},{"Start":"00:22.530 ","End":"00:24.525","Text":"cosine 0 is 1."},{"Start":"00:24.525 ","End":"00:27.300","Text":"Again, we get here 0/0,"},{"Start":"00:27.300 ","End":"00:29.355","Text":"so we can\u0027t do it by substitution."},{"Start":"00:29.355 ","End":"00:33.330","Text":"We have to use some trigonometric formulas and I\u0027ve already pre-written them,"},{"Start":"00:33.330 ","End":"00:35.925","Text":"because I see 1 minus cosine a lot,"},{"Start":"00:35.925 ","End":"00:37.610","Text":"I figured probably,"},{"Start":"00:37.610 ","End":"00:42.525","Text":"we\u0027re going to have to need this formula for 1 minus cosine of something,"},{"Start":"00:42.525 ","End":"00:46.685","Text":"say Alpha is 2 sine squared of that same thing over 2."},{"Start":"00:46.685 ","End":"00:50.480","Text":"After we do this, we\u0027re going to have a lot of sines instead of cosines,"},{"Start":"00:50.480 ","End":"00:53.575","Text":"and this is the most useful and famous formula."},{"Start":"00:53.575 ","End":"00:57.320","Text":"This is the other formula we\u0027re probably going to use."},{"Start":"00:57.320 ","End":"01:00.020","Text":"Let\u0027s continue."},{"Start":"01:00.020 ","End":"01:05.645","Text":"I\u0027m going to substitute set of Alpha 1 minus cosine x."},{"Start":"01:05.645 ","End":"01:12.390","Text":"What that will give us is the limit as x tends to"},{"Start":"01:12.390 ","End":"01:18.930","Text":"0 of 2 sine squared of this thing,"},{"Start":"01:18.930 ","End":"01:24.390","Text":"this Alpha 1 minus cosine x/2."},{"Start":"01:24.390 ","End":"01:27.825","Text":"We still have the x to the 4th here."},{"Start":"01:27.825 ","End":"01:30.800","Text":"I see there\u0027s a lot of squares here."},{"Start":"01:30.800 ","End":"01:33.920","Text":"We have sine of this thing squared and"},{"Start":"01:33.920 ","End":"01:37.930","Text":"the denominator also is a square, x squared squared."},{"Start":"01:37.930 ","End":"01:40.100","Text":"Leaving the 2 aside,"},{"Start":"01:40.100 ","End":"01:44.965","Text":"we can rewrite this as the limit,"},{"Start":"01:44.965 ","End":"01:52.325","Text":"x goes to 0 of twice something squared,"},{"Start":"01:52.325 ","End":"01:57.995","Text":"that something is going to be x squared on the denominator,"},{"Start":"01:57.995 ","End":"02:01.580","Text":"and just the sign without the squared on the numerator."},{"Start":"02:01.580 ","End":"02:10.695","Text":"It\u0027s going to be the sine of 1 minus cosine x/2."},{"Start":"02:10.695 ","End":"02:17.540","Text":"What I\u0027m going to do is because I have something squared and also there\u0027s the 2 here,"},{"Start":"02:17.540 ","End":"02:20.270","Text":"I want to do a sub exercise of just figuring"},{"Start":"02:20.270 ","End":"02:23.810","Text":"out the limit of what\u0027s inside the square brackets."},{"Start":"02:23.810 ","End":"02:32.585","Text":"What I\u0027ll do is just temporarily stop here and just work on the limit of this thing."},{"Start":"02:32.585 ","End":"02:37.360","Text":"They too will continue here what this equals after we\u0027ve done that."},{"Start":"02:37.360 ","End":"02:41.704","Text":"What I want to do is compute the limit"},{"Start":"02:41.704 ","End":"02:46.400","Text":"as x goes to 0 of what\u0027s inside the square brackets"},{"Start":"02:46.400 ","End":"02:55.625","Text":"of sine of 1 minus cosine x over 2 over x squared."},{"Start":"02:55.625 ","End":"02:59.300","Text":"What I\u0027m going to do is use this formula again,"},{"Start":"02:59.300 ","End":"03:02.240","Text":"with Alpha being just x."},{"Start":"03:02.240 ","End":"03:05.975","Text":"Now notice that I have 1 minus cosine Alpha."},{"Start":"03:05.975 ","End":"03:08.350","Text":"Here it\u0027s divided by 2,"},{"Start":"03:08.350 ","End":"03:11.960","Text":"so I could have brought the 2 over to the other side."},{"Start":"03:11.960 ","End":"03:14.420","Text":"Basically the 2 \u0027s cancel out,"},{"Start":"03:14.420 ","End":"03:23.715","Text":"so what I\u0027m left with is the limit as x goes to 0 sine of,"},{"Start":"03:23.715 ","End":"03:28.550","Text":"this thing becomes this thing without the 2 sine squared, Alpha over 2,"},{"Start":"03:28.550 ","End":"03:36.215","Text":"which is sine squared of x over 2 and still over x squared."},{"Start":"03:36.215 ","End":"03:38.390","Text":"This is going to equal,"},{"Start":"03:38.390 ","End":"03:40.520","Text":"I\u0027m going to use this formula."},{"Start":"03:40.520 ","End":"03:44.060","Text":"The 2nd 1 is sine squared of x over 2."},{"Start":"03:44.060 ","End":"03:48.295","Text":"What I\u0027d like to have is Alpha in the denominator."},{"Start":"03:48.295 ","End":"03:54.290","Text":"I\u0027ll write this as the limit x goes to 0 of"},{"Start":"03:54.290 ","End":"04:00.635","Text":"sine squared x/2."},{"Start":"04:00.635 ","End":"04:03.590","Text":"Now the sine Alpha should be over Alpha."},{"Start":"04:03.590 ","End":"04:10.130","Text":"I\u0027m going to change the denominator to sine squared x/2."},{"Start":"04:10.130 ","End":"04:14.030","Text":"But I can\u0027t just arbitrarily change the denominator,"},{"Start":"04:14.030 ","End":"04:18.020","Text":"what I have to do is to compensate by multiplying in"},{"Start":"04:18.020 ","End":"04:22.220","Text":"the numerator sine squared x/2,"},{"Start":"04:22.220 ","End":"04:24.185","Text":"so that this will cancel with this."},{"Start":"04:24.185 ","End":"04:27.290","Text":"But I still have to keep the x squared,"},{"Start":"04:27.290 ","End":"04:29.516","Text":"which is what I had originally."},{"Start":"04:29.516 ","End":"04:33.035","Text":"I\u0027ve broken this limit up into 2 separate bits."},{"Start":"04:33.035 ","End":"04:35.495","Text":"Now I\u0027d like to use this formula again,"},{"Start":"04:35.495 ","End":"04:38.465","Text":"but this time with Alpha being x/2."},{"Start":"04:38.465 ","End":"04:41.690","Text":"Now I see I have a square here and a square here."},{"Start":"04:41.690 ","End":"04:44.520","Text":"I can rewrite this,"},{"Start":"04:44.520 ","End":"04:47.550","Text":"I have to copy the whole thing again, times,"},{"Start":"04:47.550 ","End":"04:51.515","Text":"and this I can write as something squared."},{"Start":"04:51.515 ","End":"04:54.020","Text":"It\u0027s going to be something over x,"},{"Start":"04:54.020 ","End":"04:57.545","Text":"and it\u0027s going to be here sine x/2,"},{"Start":"04:57.545 ","End":"04:59.555","Text":"very close to this."},{"Start":"04:59.555 ","End":"05:03.315","Text":"How about if we just divide here by 2,"},{"Start":"05:03.315 ","End":"05:09.235","Text":"but then compensate multiplying the denominator by 2 here."},{"Start":"05:09.235 ","End":"05:12.290","Text":"That\u0027s just hasn\u0027t changed anything."},{"Start":"05:12.290 ","End":"05:20.195","Text":"This is equal to the limit x goes to 0 of this first bit."},{"Start":"05:20.195 ","End":"05:28.820","Text":"Sine of sine squared x/2 over sine squared x over"},{"Start":"05:28.820 ","End":"05:33.155","Text":"2 times sine x/2"},{"Start":"05:33.155 ","End":"05:39.345","Text":"over x over 2 squared times 1/2 squared,"},{"Start":"05:39.345 ","End":"05:44.295","Text":"which is times 1/4."},{"Start":"05:44.295 ","End":"05:49.685","Text":"We\u0027re really getting close this time because this limit,"},{"Start":"05:49.685 ","End":"05:52.260","Text":"as x goes to 0,"},{"Start":"05:52.260 ","End":"05:54.750","Text":"is the sine of Alpha over Alpha,"},{"Start":"05:54.750 ","End":"05:57.645","Text":"so this is equal to 1."},{"Start":"05:57.645 ","End":"06:01.245","Text":"This is sine Alpha over Alpha with a different Alpha."},{"Start":"06:01.245 ","End":"06:09.390","Text":"This goes also to 1 squared and the 1/4 is just a constant, stays like that."},{"Start":"06:09.390 ","End":"06:12.840","Text":"This times this, times this all together,"},{"Start":"06:12.840 ","End":"06:15.705","Text":"it\u0027s equal to 1/4."},{"Start":"06:15.705 ","End":"06:21.629","Text":"Now, this 1/4 is not the final answer."},{"Start":"06:23.360 ","End":"06:27.525","Text":"This is where we did the side exercise,"},{"Start":"06:27.525 ","End":"06:32.245","Text":"so I have to put it in to this thing in the square brackets,"},{"Start":"06:32.245 ","End":"06:36.555","Text":"I have to put this 1/4 inside the square brackets."},{"Start":"06:36.555 ","End":"06:43.740","Text":"What I get is 2 times 1/4 squared."},{"Start":"06:43.740 ","End":"06:46.215","Text":"A 1/4 was just this bit here."},{"Start":"06:46.215 ","End":"06:50.430","Text":"A 1/4 squared is 1/16 times 2,"},{"Start":"06:50.430 ","End":"06:53.235","Text":"gives us 1/8,"},{"Start":"06:53.235 ","End":"06:56.115","Text":"and this is the answer."},{"Start":"06:56.115 ","End":"06:58.390","Text":"Finally done."}],"ID":4797},{"Watched":false,"Name":"Exercise 8","Duration":"4m 30s","ChapterTopicVideoID":4790,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:04.260 ","End":"00:09.959","Text":"0 of 3 sine x minus sine 3x over x cubed."},{"Start":"00:09.959 ","End":"00:15.465","Text":"As usual, we try to substitute x equals 0 first."},{"Start":"00:15.465 ","End":"00:20.025","Text":"The denominator is 0 cubed is 0,"},{"Start":"00:20.025 ","End":"00:23.985","Text":"and the numerator sine 0 is also 0."},{"Start":"00:23.985 ","End":"00:28.725","Text":"Basically we get, if we substitute 0 over 0,"},{"Start":"00:28.725 ","End":"00:32.370","Text":"so the method of substitution doesn\u0027t quite work."},{"Start":"00:32.370 ","End":"00:35.635","Text":"We\u0027re going to have to use trigonometric formula."},{"Start":"00:35.635 ","End":"00:38.330","Text":"What I\u0027m going to do is rewrite this"},{"Start":"00:38.330 ","End":"00:40.334","Text":"in a slightly different form."},{"Start":"00:40.334 ","End":"00:47.525","Text":"I\u0027ll write this as the limit as x goes to 0 of,"},{"Start":"00:47.525 ","End":"00:49.320","Text":"instead of 3 sine x,"},{"Start":"00:49.320 ","End":"00:56.055","Text":"I\u0027m going to write 2 sine x plus another sine x,"},{"Start":"00:56.055 ","End":"00:58.200","Text":"that\u0027s altogether 3 sine x,"},{"Start":"00:58.200 ","End":"01:04.610","Text":"minus sine 3x all over x cubed."},{"Start":"01:04.610 ","End":"01:08.330","Text":"Now the reason I wrote it this way is I can make use of"},{"Start":"01:08.330 ","End":"01:12.975","Text":"a formula for the difference of 2 signs."},{"Start":"01:12.975 ","End":"01:15.130","Text":"I\u0027ll write that down."},{"Start":"01:15.130 ","End":"01:17.410","Text":"This is the formula I mean."},{"Start":"01:17.410 ","End":"01:19.365","Text":"Sine alpha minus sine beta,"},{"Start":"01:19.365 ","End":"01:22.355","Text":"and this is the expression."},{"Start":"01:22.355 ","End":"01:25.310","Text":"I also had another formula which I now am going to need"},{"Start":"01:25.310 ","End":"01:28.850","Text":"later on so I might as well write it here now."},{"Start":"01:28.850 ","End":"01:30.980","Text":"Let\u0027s continue."},{"Start":"01:30.980 ","End":"01:35.900","Text":"This equals the limit as x goes to"},{"Start":"01:35.900 ","End":"01:43.550","Text":"0 of 2 sine x, x cubed."},{"Start":"01:43.550 ","End":"01:46.190","Text":"Now, this expression I\u0027m going to use"},{"Start":"01:46.190 ","End":"01:47.330","Text":"with this formula where"},{"Start":"01:47.330 ","End":"01:51.680","Text":"alpha is going to be x and beta is going to be 3x."},{"Start":"01:51.680 ","End":"01:57.705","Text":"We get plus 2 sine half the difference,"},{"Start":"01:57.705 ","End":"01:59.680","Text":"x minus 3x,"},{"Start":"01:59.680 ","End":"02:07.570","Text":"over 2 is minus x and then times cosine of half the sum."},{"Start":"02:07.570 ","End":"02:10.830","Text":"Cosine of half the sum,"},{"Start":"02:10.830 ","End":"02:17.620","Text":"x plus 3x over 2 is 4x over 2, which is 2x."},{"Start":"02:19.190 ","End":"02:22.970","Text":"Yes, 2 sine x outside the brackets,"},{"Start":"02:22.970 ","End":"02:26.750","Text":"I have to say that sine of minus x is minus"},{"Start":"02:26.750 ","End":"02:31.795","Text":"sine x. I\u0027ll just write the whole thing again."},{"Start":"02:31.795 ","End":"02:36.815","Text":"Now, taking 2 sine x outside the brackets,"},{"Start":"02:36.815 ","End":"02:41.960","Text":"I\u0027ll get limit x goes to"},{"Start":"02:41.960 ","End":"02:48.735","Text":"0 of 2 sine x brackets,"},{"Start":"02:48.735 ","End":"02:56.505","Text":"1 minus cosine 2x over x cubed."},{"Start":"02:56.505 ","End":"02:59.660","Text":"Now you see why I wrote this formula here,"},{"Start":"02:59.660 ","End":"03:01.370","Text":"because I can write this thing in"},{"Start":"03:01.370 ","End":"03:04.865","Text":"the brackets is 2 sine squared x."},{"Start":"03:04.865 ","End":"03:10.260","Text":"We get limit as x tends to 0 of"},{"Start":"03:10.260 ","End":"03:14.805","Text":"2 sine x times"},{"Start":"03:14.805 ","End":"03:23.000","Text":"2 sine squared x all over x cubed,"},{"Start":"03:23.000 ","End":"03:27.270","Text":"which equals limit x goes to 0."},{"Start":"03:27.270 ","End":"03:29.495","Text":"The 2 with the 2 is 4,"},{"Start":"03:29.495 ","End":"03:31.810","Text":"so 4 times,"},{"Start":"03:31.810 ","End":"03:38.600","Text":"and what I\u0027m left with is sine cubed x over x cubed."},{"Start":"03:38.600 ","End":"03:44.960","Text":"This is equal to the limit as x goes"},{"Start":"03:44.960 ","End":"03:53.190","Text":"to 0 of sine x over x all cubed after the 4, yes."},{"Start":"03:53.190 ","End":"03:59.115","Text":"4 times sine x over x cubed,"},{"Start":"03:59.115 ","End":"04:01.865","Text":"and this is equal to,"},{"Start":"04:01.865 ","End":"04:05.890","Text":"since the sine x over x is a well known limit,"},{"Start":"04:05.890 ","End":"04:10.190","Text":"well, perhaps I should have written it down in the formula."},{"Start":"04:10.190 ","End":"04:12.770","Text":"This is the formula I mean,"},{"Start":"04:12.770 ","End":"04:15.310","Text":"well known and very useful."},{"Start":"04:15.310 ","End":"04:20.025","Text":"Here, this sine x over x just goes to 1."},{"Start":"04:20.025 ","End":"04:26.030","Text":"What I get is 4 times 1 cubed,"},{"Start":"04:26.030 ","End":"04:28.520","Text":"which just equals 4."},{"Start":"04:28.520 ","End":"04:30.690","Text":"That\u0027s the answer."}],"ID":4798},{"Watched":false,"Name":"Exercise 9","Duration":"5m 20s","ChapterTopicVideoID":4791,"CourseChapterTopicPlaylistID":3702,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.425","Text":"In this exercise, we have to find the limit as x tends to 0"},{"Start":"00:04.425 ","End":"00:09.300","Text":"of the function 1 minus square root of cosine x over x squared."},{"Start":"00:09.300 ","End":"00:15.255","Text":"The first thing to try is to substitute x equals 0."},{"Start":"00:15.255 ","End":"00:19.305","Text":"If x is 0, then cosine x is 1,"},{"Start":"00:19.305 ","End":"00:23.250","Text":"we get 1 minus the square root of 1 is 0, here also 0."},{"Start":"00:23.250 ","End":"00:27.960","Text":"In other words, we get 0 over 0."},{"Start":"00:27.960 ","End":"00:31.860","Text":"That\u0027s undefined, so the method of substitution won\u0027t work."},{"Start":"00:31.860 ","End":"00:34.165","Text":"We\u0027re going to have to use a different technique."},{"Start":"00:34.165 ","End":"00:37.265","Text":"Now, what I suggest is using conjugates."},{"Start":"00:37.265 ","End":"00:38.780","Text":"When you have,"},{"Start":"00:38.780 ","End":"00:40.640","Text":"in the numerator or denominator,"},{"Start":"00:40.640 ","End":"00:43.925","Text":"several terms and some with square root,"},{"Start":"00:43.925 ","End":"00:49.910","Text":"usually we use conjugates and I\u0027ll remind you what a conjugate is."},{"Start":"00:49.910 ","End":"00:55.940","Text":"If you have an arithmetical expression of the form A minus B as we do here,"},{"Start":"00:55.940 ","End":"00:58.870","Text":"and 1 or more of them is a square root,"},{"Start":"00:58.870 ","End":"01:04.425","Text":"then the conjugate is A plus B and vice versa."},{"Start":"01:04.425 ","End":"01:06.075","Text":"If we have A plus B,"},{"Start":"01:06.075 ","End":"01:08.535","Text":"its conjugate is A minus B."},{"Start":"01:08.535 ","End":"01:16.310","Text":"The reason we use conjugates is that if you multiply an expression by its conjugate,"},{"Start":"01:16.310 ","End":"01:19.310","Text":"that\u0027s the famous difference of squares formula."},{"Start":"01:19.310 ","End":"01:21.815","Text":"This is A squared minus B squared."},{"Start":"01:21.815 ","End":"01:24.095","Text":"If A and B is A square root,"},{"Start":"01:24.095 ","End":"01:27.190","Text":"when we square it, we get rid of the square root."},{"Start":"01:27.190 ","End":"01:29.280","Text":"This is going to be useful."},{"Start":"01:29.280 ","End":"01:30.840","Text":"Now, in our case,"},{"Start":"01:30.840 ","End":"01:35.075","Text":"1 is the A and the square root of cosine x is B."},{"Start":"01:35.075 ","End":"01:37.615","Text":"Let\u0027s just copy this."},{"Start":"01:37.615 ","End":"01:43.400","Text":"What we\u0027ll do is we\u0027ll multiply this by its conjugate like we did here,"},{"Start":"01:43.400 ","End":"01:48.700","Text":"multiply by 1 plus the square root of cosine x,"},{"Start":"01:48.700 ","End":"01:50.240","Text":"but you can\u0027t just multiply."},{"Start":"01:50.240 ","End":"01:52.580","Text":"It changes the exercise so you also have to divide"},{"Start":"01:52.580 ","End":"01:55.640","Text":"by the same thing or another way to look at"},{"Start":"01:55.640 ","End":"02:01.625","Text":"it is that we\u0027re multiplying by 1 because anything over itself is 1."},{"Start":"02:01.625 ","End":"02:06.125","Text":"What we get is the limit using this formula,"},{"Start":"02:06.125 ","End":"02:10.315","Text":"A minus B times A plus B is A squared minus B squared."},{"Start":"02:10.315 ","End":"02:12.690","Text":"On the numerator, we get 1 squared,"},{"Start":"02:12.690 ","End":"02:14.025","Text":"which is 1,"},{"Start":"02:14.025 ","End":"02:19.595","Text":"minus B squared is distinct squared is cosine x,"},{"Start":"02:19.595 ","End":"02:23.450","Text":"and the denominator we\u0027ll just leave as it is,"},{"Start":"02:23.450 ","End":"02:31.864","Text":"x squared times 1 plus square root of cosine x."},{"Start":"02:31.864 ","End":"02:35.330","Text":"Now, what we\u0027re going to do here is use"},{"Start":"02:35.330 ","End":"02:42.090","Text":"another formula of 1 minus cosine x, that\u0027s the conjugate."},{"Start":"02:42.090 ","End":"02:44.640","Text":"Now a bit of trig,"},{"Start":"02:44.640 ","End":"02:50.215","Text":"1 minus cosine x is equal"},{"Start":"02:50.215 ","End":"02:56.775","Text":"to 2 sine squared x over 2."},{"Start":"02:56.775 ","End":"03:00.445","Text":"The reason this is going to be useful or in general,"},{"Start":"03:00.445 ","End":"03:04.420","Text":"we expect in almost all of these trigonometrical limits to use"},{"Start":"03:04.420 ","End":"03:10.375","Text":"the famous formula that limit as any expression."},{"Start":"03:10.375 ","End":"03:19.710","Text":"Alpha goes to 0 of sine Alpha over Alpha is 1."},{"Start":"03:19.710 ","End":"03:23.000","Text":"I think we have all the formulas we need here."},{"Start":"03:23.000 ","End":"03:26.180","Text":"Now, back to the exercise."},{"Start":"03:26.180 ","End":"03:31.820","Text":"This equals limit as x goes to 0."},{"Start":"03:31.820 ","End":"03:33.980","Text":"Now, the 1 minus x,"},{"Start":"03:33.980 ","End":"03:41.205","Text":"I\u0027m going to use this formula so that 2 sine squared x over 2"},{"Start":"03:41.205 ","End":"03:49.605","Text":"over x squared 1 plus square root of cosine x."},{"Start":"03:49.605 ","End":"03:58.519","Text":"This equals, I\u0027ll write sine x over 2 over x squared."},{"Start":"03:58.519 ","End":"04:00.455","Text":"That takes care of this and this."},{"Start":"04:00.455 ","End":"04:05.630","Text":"The 2, I\u0027ll put in front here and what we\u0027re left with"},{"Start":"04:05.630 ","End":"04:14.915","Text":"is 1 over 1 plus the square root of cosine x."},{"Start":"04:14.915 ","End":"04:17.330","Text":"Now, if you look at this part,"},{"Start":"04:17.330 ","End":"04:21.380","Text":"it looks very much like this with Alpha equals x over 2,"},{"Start":"04:21.380 ","End":"04:24.800","Text":"except the denominator is x and not x over 2."},{"Start":"04:24.800 ","End":"04:28.505","Text":"How about if I just put x over 2 here,"},{"Start":"04:28.505 ","End":"04:31.805","Text":"but fix it by also writing a 2 here?"},{"Start":"04:31.805 ","End":"04:34.380","Text":"Then that should be okay."},{"Start":"04:36.880 ","End":"04:41.329","Text":"Now, if I just look at this part here,"},{"Start":"04:41.329 ","End":"04:43.685","Text":"it looks like sine Alpha over Alpha."},{"Start":"04:43.685 ","End":"04:46.235","Text":"This limit goes to 1,"},{"Start":"04:46.235 ","End":"04:50.700","Text":"so what we get is 2 from here,"},{"Start":"04:50.700 ","End":"04:54.890","Text":"1 over, the thing I circled is 1,"},{"Start":"04:54.890 ","End":"04:57.665","Text":"and here we have the 2 squared."},{"Start":"04:57.665 ","End":"04:59.060","Text":"For the last part,"},{"Start":"04:59.060 ","End":"05:01.865","Text":"I can just put x equals 0 in here."},{"Start":"05:01.865 ","End":"05:03.710","Text":"Cosine 0 is 1,"},{"Start":"05:03.710 ","End":"05:06.755","Text":"square root of 1 is 1, this is 1/2."},{"Start":"05:06.755 ","End":"05:08.935","Text":"If we do the computation,"},{"Start":"05:08.935 ","End":"05:11.535","Text":"this 2 goes with this 1/2,"},{"Start":"05:11.535 ","End":"05:13.490","Text":"cancels, 1/2 squared is 1/4."},{"Start":"05:13.490 ","End":"05:17.065","Text":"1/4, 1 over 4,"},{"Start":"05:17.065 ","End":"05:21.780","Text":"is our final answer. We\u0027re done."}],"ID":4799}],"Thumbnail":null,"ID":3702},{"Name":"Technique 8 The Sandwich Squeeze Theorem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Sandwich Theorem","Duration":"9m 35s","ChapterTopicVideoID":8252,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.015","Text":"Now we come to Technique number 8,"},{"Start":"00:03.015 ","End":"00:05.490","Text":"sometimes called The Sandwich Theorem."},{"Start":"00:05.490 ","End":"00:07.440","Text":"The theorem it\u0027s not used that often,"},{"Start":"00:07.440 ","End":"00:09.870","Text":"but sometimes comes in useful."},{"Start":"00:09.870 ","End":"00:15.015","Text":"We might have a situation where we have to compute the limit as x goes to something,"},{"Start":"00:15.015 ","End":"00:20.340","Text":"say a of f of x and we might be having some difficulty with this."},{"Start":"00:20.340 ","End":"00:25.410","Text":"But suppose we can sandwich f between 2 other functions as follows."},{"Start":"00:25.410 ","End":"00:29.515","Text":"Suppose I can find functions g of x,"},{"Start":"00:29.515 ","End":"00:33.665","Text":"which is always less than or equal to f of x and another function,"},{"Start":"00:33.665 ","End":"00:37.355","Text":"h of x, which is bigger or equal to f of x."},{"Start":"00:37.355 ","End":"00:40.850","Text":"Suppose that these 2 have limits as x goes to a."},{"Start":"00:40.850 ","End":"00:43.880","Text":"Suppose this tends to a limit, I don\u0027t know,"},{"Start":"00:43.880 ","End":"00:47.510","Text":"L. Suppose this tends to a limit L,"},{"Start":"00:47.510 ","End":"00:49.940","Text":"both of them as x goes to a,"},{"Start":"00:49.940 ","End":"00:51.950","Text":"as x goes to a."},{"Start":"00:51.950 ","End":"00:55.160","Text":"Then f being caught in between these 2"},{"Start":"00:55.160 ","End":"00:56.705","Text":"sandwich does it were."},{"Start":"00:56.705 ","End":"01:01.850","Text":"The theorem is that this will also have a limit that as x goes to a,"},{"Start":"01:01.850 ","End":"01:05.965","Text":"that will equal to L. This we know,"},{"Start":"01:05.965 ","End":"01:07.649","Text":"and this we know,"},{"Start":"01:07.649 ","End":"01:12.840","Text":"and this is what we didn\u0027t know but we can conclude from these 2 that"},{"Start":"01:12.840 ","End":"01:19.000","Text":"yes this limit also goes to L. I\u0027ll give an example and that will make it clearer."},{"Start":"01:19.000 ","End":"01:20.810","Text":"Let\u0027s say we have 3 things."},{"Start":"01:20.810 ","End":"01:23.540","Text":"We have first of all that this inequality holds in"},{"Start":"01:23.540 ","End":"01:26.450","Text":"a certain region containing a."},{"Start":"01:26.450 ","End":"01:34.880","Text":"Secondly, we have the limit as x goes to a of g of x is L. Thirdly,"},{"Start":"01:34.880 ","End":"01:43.010","Text":"we have that the limit as x goes to a of h of x equals L. If we have all these 3,"},{"Start":"01:43.010 ","End":"01:47.735","Text":"the we can conclude that the limit as x goes to"},{"Start":"01:47.735 ","End":"01:53.435","Text":"a of f of x is also equal to L. Basically,"},{"Start":"01:53.435 ","End":"01:55.534","Text":"this is the if part."},{"Start":"01:55.534 ","End":"02:00.080","Text":"If all these 3 hold, then this holds."},{"Start":"02:00.080 ","End":"02:02.450","Text":"It\u0027s not the only name it has by the way,"},{"Start":"02:02.450 ","End":"02:04.520","Text":"when f is caught between g and h,"},{"Start":"02:04.520 ","End":"02:06.770","Text":"either called the sandwich theorem."},{"Start":"02:06.770 ","End":"02:11.925","Text":"I\u0027ve also seen it called the squeeze theorem,"},{"Start":"02:11.925 ","End":"02:16.219","Text":"and I\u0027ve even seen it called the pinch theorem."},{"Start":"02:16.219 ","End":"02:20.900","Text":"Take your pick, but I\u0027ll use the notion sandwich theorem."},{"Start":"02:20.900 ","End":"02:22.580","Text":"I put this at the side,"},{"Start":"02:22.580 ","End":"02:24.289","Text":"and I\u0027m going to do an example."},{"Start":"02:24.289 ","End":"02:29.420","Text":"Let\u0027s take the limit as x goes to"},{"Start":"02:29.420 ","End":"02:35.420","Text":"infinity of 4^x plus 11^x,"},{"Start":"02:35.420 ","End":"02:39.410","Text":"all to the power of 1 over x. I want to mention"},{"Start":"02:39.410 ","End":"02:43.490","Text":"here that a could be either finite or infinite,"},{"Start":"02:43.490 ","End":"02:45.230","Text":"could have an infinite limit."},{"Start":"02:45.230 ","End":"02:47.570","Text":"It would also same theorem works and here we have"},{"Start":"02:47.570 ","End":"02:51.220","Text":"an example where our limit goes to infinity."},{"Start":"02:51.220 ","End":"02:54.920","Text":"We can\u0027t do this limit directly because when x goes to infinity,"},{"Start":"02:54.920 ","End":"03:00.620","Text":"this goes to infinity and this goes to infinity but 1 over infinity goes to 0,"},{"Start":"03:00.620 ","End":"03:03.230","Text":"so we have an infinity to the power of 0,"},{"Start":"03:03.230 ","End":"03:06.380","Text":"a situation that\u0027s not defined."},{"Start":"03:06.380 ","End":"03:08.600","Text":"We\u0027re going to have to use some technique and of course,"},{"Start":"03:08.600 ","End":"03:10.310","Text":"we\u0027re going to use the sandwich theorem."},{"Start":"03:10.310 ","End":"03:13.885","Text":"What we do is this is going to be our f of x"},{"Start":"03:13.885 ","End":"03:17.950","Text":"and we want to try and get it sandwiched between some g and h,"},{"Start":"03:17.950 ","End":"03:21.760","Text":"where hopefully g and h have limited or easier to compute."},{"Start":"03:21.760 ","End":"03:24.115","Text":"Let\u0027s see how we do this."},{"Start":"03:24.115 ","End":"03:26.615","Text":"Let\u0027s first of all just take this part."},{"Start":"03:26.615 ","End":"03:29.290","Text":"4^x plus 11^x."},{"Start":"03:29.290 ","End":"03:33.920","Text":"We\u0027re assuming x is positive because it\u0027s going to plus infinity."},{"Start":"03:33.930 ","End":"03:38.775","Text":"4^x is less than 11^x whenever x is positive."},{"Start":"03:38.775 ","End":"03:41.700","Text":"This thing I\u0027ll copy just as is."},{"Start":"03:41.700 ","End":"03:43.465","Text":"That\u0027s 1 side."},{"Start":"03:43.465 ","End":"03:48.430","Text":"On the other hand, this is bigger than just 11^x on its own."},{"Start":"03:48.430 ","End":"03:51.220","Text":"This is certainly less than this because it\u0027s got a bit extra."},{"Start":"03:51.220 ","End":"03:54.680","Text":"When we have less than just to conform with the theorem,"},{"Start":"03:54.680 ","End":"03:58.580","Text":"I\u0027ll write it as less than or equal to it\u0027s certainly okay to do that."},{"Start":"03:58.580 ","End":"04:02.330","Text":"Also, I can take everything to the power of 1 over x,"},{"Start":"04:02.330 ","End":"04:03.680","Text":"as long as x is positive,"},{"Start":"04:03.680 ","End":"04:05.960","Text":"it is positive, it is going to infinity."},{"Start":"04:05.960 ","End":"04:08.585","Text":"Here to the power of 1 over x,"},{"Start":"04:08.585 ","End":"04:11.210","Text":"and here to the power of 1 over x."},{"Start":"04:11.210 ","End":"04:12.690","Text":"It\u0027s a standard trick,"},{"Start":"04:12.690 ","End":"04:16.160","Text":"so you wouldn\u0027t necessarily see right away why I\u0027m doing what I\u0027m doing,"},{"Start":"04:16.160 ","End":"04:18.610","Text":"but only you follow that this is so."},{"Start":"04:18.610 ","End":"04:21.050","Text":"We have this double inequality."},{"Start":"04:21.050 ","End":"04:24.965","Text":"Now let\u0027s see if we can find the limits of these 2."},{"Start":"04:24.965 ","End":"04:26.990","Text":"Hopefully, not only do they exist,"},{"Start":"04:26.990 ","End":"04:29.360","Text":"but they come out the same. Let\u0027s see."},{"Start":"04:29.360 ","End":"04:31.445","Text":"Let\u0027s take the first 1, the easier 1."},{"Start":"04:31.445 ","End":"04:39.005","Text":"The limit as x goes to infinity of 11^x to the 1 over x."},{"Start":"04:39.005 ","End":"04:41.600","Text":"This using the exponent rules,"},{"Start":"04:41.600 ","End":"04:43.770","Text":"x times 1 over x is 1."},{"Start":"04:43.770 ","End":"04:50.270","Text":"This is just the limit as x goes to infinity of 11 and it\u0027s just a constant 11."},{"Start":"04:50.270 ","End":"04:55.370","Text":"If this was my function g and this is the function h,"},{"Start":"04:55.370 ","End":"04:57.170","Text":"and this is our original function f,"},{"Start":"04:57.170 ","End":"05:00.754","Text":"then g goes to 11."},{"Start":"05:00.754 ","End":"05:04.100","Text":"Now let\u0027s see about the h, the right-hand 1."},{"Start":"05:04.100 ","End":"05:07.715","Text":"Here we have the limit as x goes to infinity."},{"Start":"05:07.715 ","End":"05:11.060","Text":"Now this thing is twice 11^x."},{"Start":"05:11.060 ","End":"05:13.775","Text":"Something plus itself is twice that something,"},{"Start":"05:13.775 ","End":"05:19.160","Text":"twice 11^x, to the power of 1 over x."},{"Start":"05:19.160 ","End":"05:21.545","Text":"Now I can take the 1 over x,"},{"Start":"05:21.545 ","End":"05:23.630","Text":"the exponent on each bit separately."},{"Start":"05:23.630 ","End":"05:33.260","Text":"I can say it\u0027s the limit as x goes to infinity of 2^1/x times this already"},{"Start":"05:33.260 ","End":"05:35.450","Text":"we\u0027ve seen above is 11."},{"Start":"05:35.450 ","End":"05:37.580","Text":"11 is a constant,"},{"Start":"05:37.580 ","End":"05:43.150","Text":"and this bit 2^1/x goes to 1."},{"Start":"05:43.150 ","End":"05:46.925","Text":"Basically you could say that this is equal to as x goes to infinity,"},{"Start":"05:46.925 ","End":"05:48.860","Text":"1 over x goes to 0."},{"Start":"05:48.860 ","End":"05:51.919","Text":"We have 2^0 times 11,"},{"Start":"05:51.919 ","End":"05:54.110","Text":"and 2^0 is 1."},{"Start":"05:54.110 ","End":"05:56.915","Text":"This is just equal to 11."},{"Start":"05:56.915 ","End":"06:02.825","Text":"We have that this part where g tends to 11,"},{"Start":"06:02.825 ","End":"06:08.240","Text":"and we got that h tends to 11 as x goes to infinity and so on."},{"Start":"06:08.240 ","End":"06:14.870","Text":"We can conclude that the limit as x goes to infinity of our function,"},{"Start":"06:14.870 ","End":"06:16.955","Text":"which is f of x,"},{"Start":"06:16.955 ","End":"06:19.720","Text":"is also equal to 11,"},{"Start":"06:19.720 ","End":"06:21.405","Text":"and that\u0027s the answer."},{"Start":"06:21.405 ","End":"06:26.990","Text":"The second example I\u0027m going to bring is the limit as x goes to"},{"Start":"06:26.990 ","End":"06:33.080","Text":"0 of x times the sine of 1 over x."},{"Start":"06:33.080 ","End":"06:35.435","Text":"Now, why is this a problem?"},{"Start":"06:35.435 ","End":"06:38.750","Text":"When x goes to 0, the first part goes to 0,"},{"Start":"06:38.750 ","End":"06:41.210","Text":"but what about the sine of 1 over x?"},{"Start":"06:41.210 ","End":"06:44.630","Text":"Well, when x goes to 0, there is no limit of 1 over x."},{"Start":"06:44.630 ","End":"06:49.025","Text":"But even if I take a one-sided limit let\u0027s say x goes to 0 from the right,"},{"Start":"06:49.025 ","End":"06:53.315","Text":"1 over x would go to infinity and there is no sign of infinity."},{"Start":"06:53.315 ","End":"06:55.040","Text":"When something goes to infinity,"},{"Start":"06:55.040 ","End":"07:00.215","Text":"the sign keeps oscillating from plus 1 to minus 1 to plus 1 to minus 1."},{"Start":"07:00.215 ","End":"07:02.090","Text":"When something goes to infinity,"},{"Start":"07:02.090 ","End":"07:04.310","Text":"there is no limit to the sign,"},{"Start":"07:04.310 ","End":"07:11.530","Text":"and I don\u0027t know that we have a case for 0 times non-existent,"},{"Start":"07:11.530 ","End":"07:13.705","Text":"that\u0027s basically what we have here."},{"Start":"07:13.705 ","End":"07:16.965","Text":"We can\u0027t say that 0 times non-existent is 0,"},{"Start":"07:16.965 ","End":"07:18.710","Text":"there are conditions when you can,"},{"Start":"07:18.710 ","End":"07:20.390","Text":"but I don\u0027t want get into that."},{"Start":"07:20.390 ","End":"07:23.465","Text":"This is a classic case for the sandwich theorem."},{"Start":"07:23.465 ","End":"07:28.255","Text":"Remember I said something about this oscillating between 1 and minus 1."},{"Start":"07:28.255 ","End":"07:31.970","Text":"That\u0027s the key because you see the sine of anything,"},{"Start":"07:31.970 ","End":"07:39.340","Text":"the sine of any angle Alpha is always between minus 1 and 1."},{"Start":"07:39.340 ","End":"07:42.050","Text":"Because of this trigonometric property,"},{"Start":"07:42.050 ","End":"07:48.620","Text":"what we can do is then say that x sine of 1 over x is going to"},{"Start":"07:48.620 ","End":"07:56.405","Text":"be less than x times 1 and bigger or equal to x times minus 1."},{"Start":"07:56.405 ","End":"07:59.990","Text":"But there\u0027s something not quite right with this because"},{"Start":"07:59.990 ","End":"08:03.620","Text":"I can\u0027t multiply this inequality by a negative value x."},{"Start":"08:03.620 ","End":"08:04.910","Text":"If x was positive,"},{"Start":"08:04.910 ","End":"08:07.520","Text":"I\u0027d be okay, but it could be negative."},{"Start":"08:07.520 ","End":"08:10.370","Text":"What I\u0027m going to do is write here the absolute value of x."},{"Start":"08:10.370 ","End":"08:13.089","Text":"This will take care of positive and negative."},{"Start":"08:13.089 ","End":"08:16.550","Text":"Multiply a number by something between plus or minus 1,"},{"Start":"08:16.550 ","End":"08:18.620","Text":"it\u0027s most going to be the absolute value of"},{"Start":"08:18.620 ","End":"08:22.025","Text":"that number or minus the absolute value of that number."},{"Start":"08:22.025 ","End":"08:25.730","Text":"This is going to be our function g from the theorem,"},{"Start":"08:25.730 ","End":"08:28.210","Text":"this is going to be our f of x."},{"Start":"08:28.210 ","End":"08:30.885","Text":"This is going to be our h of x."},{"Start":"08:30.885 ","End":"08:37.970","Text":"Then the limit as x goes to 0 of minus absolute value of x."},{"Start":"08:37.970 ","End":"08:41.585","Text":"You substitute x equals 0 is minus 0 is 0,"},{"Start":"08:41.585 ","End":"08:47.630","Text":"and the limit as x goes to 0 of the absolute value of x is also 0."},{"Start":"08:47.630 ","End":"08:50.345","Text":"Therefore, from the theorem,"},{"Start":"08:50.345 ","End":"09:00.620","Text":"we can conclude that the limit as x goes to 0 of x sine of 1 over x is also equal to 0."},{"Start":"09:00.620 ","End":"09:03.110","Text":"0 is our L in this case,"},{"Start":"09:03.110 ","End":"09:04.595","Text":"and that\u0027s the answer."},{"Start":"09:04.595 ","End":"09:09.920","Text":"But I\u0027ll just mention something I started to match here about 0 times non-existent."},{"Start":"09:09.920 ","End":"09:13.100","Text":"But if I looked at it as 0 times bounded,"},{"Start":"09:13.100 ","End":"09:17.329","Text":"there is a theorem and something goes to 0 and something else is bounded,"},{"Start":"09:17.329 ","End":"09:19.580","Text":"sine is always between 1 and minus 1,"},{"Start":"09:19.580 ","End":"09:22.190","Text":"then the limit also goes to 0."},{"Start":"09:22.190 ","End":"09:28.565","Text":"If you had loved the theorem that something goes to 0 times something bounded,"},{"Start":"09:28.565 ","End":"09:31.325","Text":"then we could have saved using the sandwich theorem,"},{"Start":"09:31.325 ","End":"09:35.580","Text":"but I wanted to use it for the practice. We\u0027re done."}],"ID":8412},{"Watched":false,"Name":"Exercise 1","Duration":"3m 23s","ChapterTopicVideoID":1570,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:08.985","Text":"In this exercise, we have to find the limit as x goes to infinity of sine x over x."},{"Start":"00:08.985 ","End":"00:11.715","Text":"We\u0027re very familiar with this,"},{"Start":"00:11.715 ","End":"00:13.665","Text":"but not quite in this form."},{"Start":"00:13.665 ","End":"00:15.840","Text":"We know that if x goes to 0,"},{"Start":"00:15.840 ","End":"00:18.675","Text":"sine x over x goes to 1."},{"Start":"00:18.675 ","End":"00:21.195","Text":"But here we have infinity, notice."},{"Start":"00:21.195 ","End":"00:29.460","Text":"What we\u0027re going to do is use some law that comes from the Sandwich law."},{"Start":"00:29.460 ","End":"00:31.650","Text":"But we just have to know it as is."},{"Start":"00:31.650 ","End":"00:34.470","Text":"That is, that if f of x is bounded,"},{"Start":"00:34.470 ","End":"00:36.734","Text":"and g of x tends to 0,"},{"Start":"00:36.734 ","End":"00:38.564","Text":"these are 2 functions."},{"Start":"00:38.564 ","End":"00:41.850","Text":"Then f of x, g of x tends to 0."},{"Start":"00:41.850 ","End":"00:43.590","Text":"Assuming the limit is the same,"},{"Start":"00:43.590 ","End":"00:45.960","Text":"if x goes to a here,"},{"Start":"00:45.960 ","End":"00:48.080","Text":"and also to a here, then f of x,"},{"Start":"00:48.080 ","End":"00:51.965","Text":"g of x as x goes to a also tends to 0."},{"Start":"00:51.965 ","End":"00:57.815","Text":"We\u0027re going to be coming back to this a few times so it\u0027s worth remembering it."},{"Start":"00:57.815 ","End":"01:01.110","Text":"What we do is using this rule,"},{"Start":"01:08.030 ","End":"01:16.320","Text":"put identical f of x will take as sine x."},{"Start":"01:18.970 ","End":"01:27.840","Text":"We let g of x be the same thing as 1 over x."},{"Start":"01:27.840 ","End":"01:32.050","Text":"We take the limit to be infinity,"},{"Start":"01:32.240 ","End":"01:37.080","Text":"then sine x is bounded."},{"Start":"01:37.080 ","End":"01:40.770","Text":"Now, why is sine x bounded?"},{"Start":"01:40.770 ","End":"01:44.795","Text":"Because it\u0027s always between minus 1, and 1."},{"Start":"01:44.795 ","End":"01:47.330","Text":"I could write this to remind you"},{"Start":"01:47.330 ","End":"01:54.125","Text":"that always minus 1 is less than or equal to sine of anything,"},{"Start":"01:54.125 ","End":"01:56.255","Text":"and that\u0027s less than or equal to 1."},{"Start":"01:56.255 ","End":"01:58.220","Text":"Bounded between minus 1,"},{"Start":"01:58.220 ","End":"02:01.400","Text":"and 1, and g of x."},{"Start":"02:01.400 ","End":"02:04.180","Text":"Why does it go to 0?"},{"Start":"02:04.180 ","End":"02:07.125","Text":"Well, when x goes to infinity,"},{"Start":"02:07.125 ","End":"02:10.070","Text":"we get 1 over infinity, which is 0."},{"Start":"02:10.070 ","End":"02:15.780","Text":"This 1 tends to 0."},{"Start":"02:15.780 ","End":"02:18.185","Text":"Now we\u0027ve met the first condition for f,"},{"Start":"02:18.185 ","End":"02:23.190","Text":"the second condition for g. The product also tends to 0."},{"Start":"02:23.290 ","End":"02:32.360","Text":"From these 2, it follows as a corollary that f of"},{"Start":"02:32.360 ","End":"02:40.590","Text":"x times g of x also goes to 0."},{"Start":"02:40.590 ","End":"02:45.020","Text":"But f of x, g of x is just sine of x 1 over x,"},{"Start":"02:45.020 ","End":"02:47.585","Text":"which means that, in other words,"},{"Start":"02:47.585 ","End":"02:53.565","Text":"sine x over x also goes to 0."},{"Start":"02:53.565 ","End":"02:56.285","Text":"I often omit the word limit,"},{"Start":"02:56.285 ","End":"02:57.620","Text":"because for something tends to,"},{"Start":"02:57.620 ","End":"02:59.660","Text":"something means it\u0027s a limit,"},{"Start":"02:59.660 ","End":"03:02.270","Text":"and it\u0027s obvious what the limit is,"},{"Start":"03:02.270 ","End":"03:03.935","Text":"then I could write here,"},{"Start":"03:03.935 ","End":"03:07.010","Text":"if I want to put in here to be very precise."},{"Start":"03:07.010 ","End":"03:12.235","Text":"I will put the limit as x goes to infinity."},{"Start":"03:12.235 ","End":"03:15.950","Text":"Sometimes, we mathematicians take some shortcuts."},{"Start":"03:15.950 ","End":"03:18.485","Text":"Okay, anyway, that\u0027ll do as a proof,"},{"Start":"03:18.485 ","End":"03:23.010","Text":"just straightforward application of this principle, and we\u0027re done."}],"ID":1582},{"Watched":false,"Name":"Exercise 2","Duration":"2m 33s","ChapterTopicVideoID":1571,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.710","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression."},{"Start":"00:05.710 ","End":"00:10.745","Text":"Let\u0027s see what happens if we put x equals infinity in here."},{"Start":"00:10.745 ","End":"00:15.000","Text":"In the denominator, we get infinity."},{"Start":"00:15.000 ","End":"00:17.385","Text":"In the numerator,"},{"Start":"00:17.385 ","End":"00:22.195","Text":"we get cosine of something that goes to infinity, and what\u0027s that?"},{"Start":"00:22.195 ","End":"00:28.300","Text":"Well, actually, this has no limit because it keeps oscillating between 1 and minus 1,"},{"Start":"00:28.300 ","End":"00:30.010","Text":"1 minus 1, and so on."},{"Start":"00:30.010 ","End":"00:31.715","Text":"It just doesn\u0027t have a limit."},{"Start":"00:31.715 ","End":"00:35.125","Text":"That\u0027s where this theorem comes to help us out."},{"Start":"00:35.125 ","End":"00:39.835","Text":"If we have one function which is bounded and another one which tends to 0,"},{"Start":"00:39.835 ","End":"00:42.660","Text":"then their product goes to 0."},{"Start":"00:42.660 ","End":"00:44.910","Text":"That\u0027s what this says in simple words."},{"Start":"00:44.910 ","End":"00:49.890","Text":"Something that goes to 0 times something bounded, you get 0."},{"Start":"00:50.060 ","End":"00:54.015","Text":"Where do we see a product in our case?"},{"Start":"00:54.015 ","End":"00:57.200","Text":"There\u0027s a quotient. If you slightly rewrite it"},{"Start":"00:57.200 ","End":"01:01.085","Text":"as the numerator times 1 over x, that ought to do it."},{"Start":"01:01.085 ","End":"01:05.640","Text":"Let\u0027s take f of x."},{"Start":"01:05.640 ","End":"01:07.545","Text":"That\u0027s the bounded one."},{"Start":"01:07.545 ","End":"01:13.965","Text":"We\u0027ll take f of x is cosine of 2x plus 1."},{"Start":"01:13.965 ","End":"01:15.615","Text":"We\u0027ll take the other one,"},{"Start":"01:15.615 ","End":"01:20.150","Text":"g of x is equal to 1 over x."},{"Start":"01:20.150 ","End":"01:26.100","Text":"Now, this one certainly satisfies the condition that it\u0027s bounded."},{"Start":"01:26.750 ","End":"01:32.125","Text":"In simple words, this means that it can\u0027t go above or below a certain level."},{"Start":"01:32.125 ","End":"01:36.160","Text":"It stays between minus 1 and 1, so it\u0027s bounded."},{"Start":"01:36.160 ","End":"01:40.510","Text":"The other one has to tend to 0 or go to 0."},{"Start":"01:40.510 ","End":"01:41.980","Text":"When x goes to infinity,"},{"Start":"01:41.980 ","End":"01:44.660","Text":"this goes to 0."},{"Start":"01:48.920 ","End":"01:51.360","Text":"We have an f which is bounded,"},{"Start":"01:51.360 ","End":"01:52.470","Text":"a g that goes to 0,"},{"Start":"01:52.470 ","End":"01:55.440","Text":"so the product f of x g of x,"},{"Start":"01:55.440 ","End":"01:59.055","Text":"and tends to means that\u0027s what the limit is."},{"Start":"01:59.055 ","End":"02:03.730","Text":"So the limit as x goes to"},{"Start":"02:03.730 ","End":"02:13.445","Text":"infinity of f of x times g of x is equal to 0."},{"Start":"02:13.445 ","End":"02:15.080","Text":"But what is f of x,"},{"Start":"02:15.080 ","End":"02:17.900","Text":"g of x, f of x times g of x,"},{"Start":"02:17.900 ","End":"02:25.935","Text":"is just cosine of 2x minus 1 over x. I needn\u0027t write that again."},{"Start":"02:25.935 ","End":"02:29.775","Text":"That\u0027s should do for finding the limit."},{"Start":"02:29.775 ","End":"02:32.050","Text":"The answer is 0."}],"ID":1583},{"Watched":false,"Name":"Exercise 3","Duration":"5m 7s","ChapterTopicVideoID":1572,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:07.080","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression."},{"Start":"00:07.080 ","End":"00:12.060","Text":"If we put x equals infinity in the numerator,"},{"Start":"00:12.060 ","End":"00:17.610","Text":"this thing goes to infinity plus something that\u0027s between 1 and minus 1."},{"Start":"00:17.610 ","End":"00:18.810","Text":"It goes to infinity,"},{"Start":"00:18.810 ","End":"00:21.810","Text":"the same way the denominator also goes to infinity."},{"Start":"00:21.810 ","End":"00:27.930","Text":"We have infinity over infinity and that\u0027s undefined."},{"Start":"00:27.930 ","End":"00:30.990","Text":"We\u0027re going to use some trick."},{"Start":"00:30.990 ","End":"00:38.900","Text":"What I\u0027ve done is written here 1 of these laws or theorems from calculus,"},{"Start":"00:38.900 ","End":"00:41.750","Text":"that if we have 2 functions, f and g,"},{"Start":"00:41.750 ","End":"00:45.725","Text":"1 of them is bounded and the other 1 goes to 0,"},{"Start":"00:45.725 ","End":"00:50.195","Text":"then the product function also goes to 0."},{"Start":"00:50.195 ","End":"00:54.130","Text":"This is used a lot and it\u0027s worth remembering."},{"Start":"00:54.130 ","End":"00:57.080","Text":"Let\u0027s see how we\u0027re going to use this here."},{"Start":"00:57.080 ","End":"01:01.055","Text":"It isn\u0027t immediately apparent how to break it up, but look."},{"Start":"01:01.055 ","End":"01:03.710","Text":"If we take x out of the numerator,"},{"Start":"01:03.710 ","End":"01:06.755","Text":"then something good might happen. Let\u0027s see."},{"Start":"01:06.755 ","End":"01:08.465","Text":"If we take this limit,"},{"Start":"01:08.465 ","End":"01:17.205","Text":"then this equals the limit as x goes to infinity."},{"Start":"01:17.205 ","End":"01:21.780","Text":"Now, we take x outside the brackets, like we often do."},{"Start":"01:21.780 ","End":"01:30.510","Text":"What we\u0027re left with is 3 plus sine x over x"},{"Start":"01:30.510 ","End":"01:36.090","Text":"divided by x brackets"},{"Start":"01:36.090 ","End":"01:43.240","Text":"4 plus cosine x over x."},{"Start":"01:44.330 ","End":"01:47.880","Text":"Now, the x cancels because it\u0027s not 0,"},{"Start":"01:47.880 ","End":"01:53.020","Text":"it goes to infinity and what we\u0027re left with is a fraction."},{"Start":"01:55.850 ","End":"02:00.495","Text":"In fact if we just look at the sine x over x part,"},{"Start":"02:00.495 ","End":"02:07.150","Text":"let\u0027s just put this as an asterisk and let\u0027s put this as a double asterisk,"},{"Start":"02:10.280 ","End":"02:13.965","Text":"1 asterisk, 2 asterisks,"},{"Start":"02:13.965 ","End":"02:18.810","Text":"then what we get for asterisk is"},{"Start":"02:18.810 ","End":"02:28.605","Text":"the limit as x goes to infinity of sine x over x."},{"Start":"02:28.605 ","End":"02:33.575","Text":"That equals, we\u0027ve done this 1 before, it equals 0."},{"Start":"02:33.575 ","End":"02:36.275","Text":"But in case you don\u0027t remember,"},{"Start":"02:36.275 ","End":"02:41.910","Text":"then what we do is we write the sine x over x."},{"Start":"02:44.020 ","End":"02:48.049","Text":"We write it as 1 over x."},{"Start":"02:48.049 ","End":"02:50.195","Text":"Well, let\u0027s do it the other way round."},{"Start":"02:50.195 ","End":"02:52.445","Text":"We want 1 of the first 1 to be bounded,"},{"Start":"02:52.445 ","End":"02:57.110","Text":"that sine x times 1 over x."},{"Start":"02:57.110 ","End":"03:01.910","Text":"Now, if I look at the sine x and call this 1 f and I call the 1"},{"Start":"03:01.910 ","End":"03:07.285","Text":"over x as g and what we have originally is fg,"},{"Start":"03:07.285 ","End":"03:11.810","Text":"then it says that if f is bounded and indeed sine x is"},{"Start":"03:11.810 ","End":"03:18.450","Text":"bounded between minus 1 and 1, so that\u0027s bounded."},{"Start":"03:19.010 ","End":"03:23.110","Text":"We have something that\u0027s bounded,"},{"Start":"03:25.450 ","End":"03:33.725","Text":"that\u0027s the f. The 1 over x part goes to 0."},{"Start":"03:33.725 ","End":"03:37.590","Text":"Let\u0027s just write goes to 0,"},{"Start":"03:37.590 ","End":"03:39.645","Text":"it\u0027s shorter than tends,"},{"Start":"03:39.645 ","End":"03:47.204","Text":"goes to 0, that\u0027s the g. It fits all the conditions."},{"Start":"03:47.204 ","End":"03:53.040","Text":"If f is bounded and g goes to 0,"},{"Start":"03:53.040 ","End":"03:55.095","Text":"then fg goes to 0."},{"Start":"03:55.095 ","End":"03:57.210","Text":"That justifies this."},{"Start":"03:57.210 ","End":"04:06.670","Text":"Now, the double asterisk says that,"},{"Start":"04:07.550 ","End":"04:12.405","Text":"just scroll up a bit here to give us some more room,"},{"Start":"04:12.405 ","End":"04:20.330","Text":"the double asterisk says that the limit is"},{"Start":"04:20.330 ","End":"04:29.540","Text":"the limit of x goes to infinity of cosine x over x."},{"Start":"04:29.540 ","End":"04:33.770","Text":"Now in exactly the same way as for sine x, what\u0027s the difference?"},{"Start":"04:33.770 ","End":"04:38.250","Text":"Sine x is bounded between minus 1 and 1 and so is the cosine."},{"Start":"04:38.250 ","End":"04:40.340","Text":"In a very similar way,"},{"Start":"04:40.340 ","End":"04:44.150","Text":"we don\u0027t have to do all the work again, this equals 0."},{"Start":"04:44.150 ","End":"04:47.000","Text":"Now that we\u0027ve done asterisk and asterisk,"},{"Start":"04:47.000 ","End":"04:50.130","Text":"our final answer will be"},{"Start":"04:50.500 ","End":"04:59.705","Text":"3 plus 0 over 4 plus 0."},{"Start":"04:59.705 ","End":"05:05.300","Text":"This is equal to 3/4."},{"Start":"05:05.300 ","End":"05:07.380","Text":"That\u0027s the answer."}],"ID":1584},{"Watched":false,"Name":"Exercise 4","Duration":"4m 44s","ChapterTopicVideoID":1573,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In this exercise,"},{"Start":"00:01.980 ","End":"00:07.035","Text":"we have to find the limit as x goes to infinity of this expression,"},{"Start":"00:07.035 ","End":"00:10.905","Text":"which has some trigonometric stuff in it."},{"Start":"00:10.905 ","End":"00:13.920","Text":"If we try to substitute x equals infinity,"},{"Start":"00:13.920 ","End":"00:15.315","Text":"which is the first thing we try,"},{"Start":"00:15.315 ","End":"00:18.510","Text":"we get plus infinity plus infinity."},{"Start":"00:18.510 ","End":"00:22.020","Text":"But sine of 2x doesn\u0027t have a limit."},{"Start":"00:22.020 ","End":"00:23.265","Text":"It just keeps going,"},{"Start":"00:23.265 ","End":"00:26.085","Text":"oscillating from 1 to minus 1 and so on."},{"Start":"00:26.085 ","End":"00:27.765","Text":"Similarly, in denominator,"},{"Start":"00:27.765 ","End":"00:29.535","Text":"this goes to infinity,"},{"Start":"00:29.535 ","End":"00:33.555","Text":"but here we have something which doesn\u0027t have a limit and oscillates."},{"Start":"00:33.555 ","End":"00:37.410","Text":"Even if it was infinity over infinity that still wouldn\u0027t help us,"},{"Start":"00:37.410 ","End":"00:39.260","Text":"so we\u0027re going to have to use some tricks."},{"Start":"00:39.260 ","End":"00:42.530","Text":"The 1 I\u0027m going to use here is 1"},{"Start":"00:42.530 ","End":"00:46.165","Text":"of these rules or laws that follows from the sandwich theorem,"},{"Start":"00:46.165 ","End":"00:48.120","Text":"that if we have 2 functions,"},{"Start":"00:48.120 ","End":"00:53.900","Text":"f and g, 1 unbounded and the other 1 tends to 0 and we take their product,"},{"Start":"00:53.900 ","End":"00:56.075","Text":"then it will also go to 0,"},{"Start":"00:56.075 ","End":"00:57.725","Text":"just like the second 1 does."},{"Start":"00:57.725 ","End":"01:02.050","Text":"In other words, bounded times going to 0 equals going to 0."},{"Start":"01:02.050 ","End":"01:06.050","Text":"Isn\u0027t immediately obvious how to break this thing up,"},{"Start":"01:06.050 ","End":"01:08.990","Text":"but we\u0027ll do a little bit of algebra first and"},{"Start":"01:08.990 ","End":"01:12.470","Text":"take out the highest powers as we often do."},{"Start":"01:12.470 ","End":"01:15.710","Text":"This thing is equal to"},{"Start":"01:15.710 ","End":"01:23.435","Text":"the limit as x goes to infinity."},{"Start":"01:23.435 ","End":"01:25.880","Text":"I\u0027m taking x squared out of here,"},{"Start":"01:25.880 ","End":"01:33.585","Text":"so it\u0027s x squared times 3 plus 1/x"},{"Start":"01:33.585 ","End":"01:38.120","Text":"plus 1/x squared"},{"Start":"01:38.120 ","End":"01:44.055","Text":"sine 2x over,"},{"Start":"01:44.055 ","End":"01:46.815","Text":"and here we also have x squared,"},{"Start":"01:46.815 ","End":"01:51.090","Text":"x squared times 1 plus"},{"Start":"01:51.090 ","End":"01:58.050","Text":"1/x squared cosine 3x."},{"Start":"01:58.050 ","End":"02:03.720","Text":"The good thing about this is that we\u0027re now able to cancel the x squared."},{"Start":"02:04.630 ","End":"02:11.675","Text":"Now, if we look at the numerator and denominator separately,"},{"Start":"02:11.675 ","End":"02:13.280","Text":"In the numerator,"},{"Start":"02:13.280 ","End":"02:15.515","Text":"this part goes to 3,"},{"Start":"02:15.515 ","End":"02:19.785","Text":"this part goes to 0, here we have 1,"},{"Start":"02:19.785 ","End":"02:22.140","Text":"and here, and here,"},{"Start":"02:22.140 ","End":"02:24.935","Text":"in both these places we have a product,"},{"Start":"02:24.935 ","End":"02:27.005","Text":"1/x squared times something."},{"Start":"02:27.005 ","End":"02:33.840","Text":"For these 2, the top 1 we\u0027ll call it asterisk and the bottom 1,"},{"Start":"02:33.840 ","End":"02:36.365","Text":"and I mean this piece here,"},{"Start":"02:36.365 ","End":"02:38.720","Text":"and in the bottom part,"},{"Start":"02:38.720 ","End":"02:43.690","Text":"this piece here, I\u0027ll call it that say double asterisk."},{"Start":"02:43.690 ","End":"02:46.399","Text":"If I eat to each of these separately,"},{"Start":"02:46.399 ","End":"02:49.760","Text":"then I\u0027ll be able to substitute back in here and get the answer."},{"Start":"02:49.760 ","End":"02:52.490","Text":"Now if we take the first 1 ,"},{"Start":"02:52.490 ","End":"03:00.060","Text":"if we look at 1/x squared times sine of 2x,"},{"Start":"03:00.060 ","End":"03:02.730","Text":"and 1 of them is got to be bounded,"},{"Start":"03:02.730 ","End":"03:04.080","Text":"one of them tends to 0,"},{"Start":"03:04.080 ","End":"03:09.855","Text":"so let\u0027s say that this 1 is g and this 1 is f,"},{"Start":"03:09.855 ","End":"03:20.045","Text":"then certainly g goes to 0 as x goes to infinity and sine of 2x is bounded."},{"Start":"03:20.045 ","End":"03:24.170","Text":"As I mentioned, it\u0027s bound between minus 1 and plus 1."},{"Start":"03:24.170 ","End":"03:25.850","Text":"It can never get outside those values."},{"Start":"03:25.850 ","End":"03:28.790","Text":"Here we have bounded times going to 0,"},{"Start":"03:28.790 ","End":"03:39.080","Text":"so asterisk, which is the product is equal to 0."},{"Start":"03:39.080 ","End":"03:40.610","Text":"Let\u0027s just write it that way."},{"Start":"03:40.610 ","End":"03:43.250","Text":"Asterisk is the limit of this thing."},{"Start":"03:43.250 ","End":"03:52.450","Text":"Similarly, when we have the 1/x squared times cosine 3x,"},{"Start":"03:52.610 ","End":"03:56.015","Text":"there\u0027s no essential difference here."},{"Start":"03:56.015 ","End":"03:57.080","Text":"This goes to 0,"},{"Start":"03:57.080 ","End":"04:00.725","Text":"this is bounded also by plus and minus 1."},{"Start":"04:00.725 ","End":"04:05.270","Text":"Our double star also equals,"},{"Start":"04:05.270 ","End":"04:08.910","Text":"but we\u0027re going to plug in there."},{"Start":"04:08.910 ","End":"04:12.920","Text":"Altogether, after we\u0027ve done these 2 separate bits,"},{"Start":"04:12.920 ","End":"04:19.320","Text":"what we get here is 3 plus, well,"},{"Start":"04:19.320 ","End":"04:22.630","Text":"we can put this 1 into plus 0,"},{"Start":"04:23.180 ","End":"04:27.070","Text":"and this thing is also 0,"},{"Start":"04:27.710 ","End":"04:31.335","Text":"divided by 1 plus,"},{"Start":"04:31.335 ","End":"04:35.460","Text":"and again, these 2 asterisks came out to be 0."},{"Start":"04:35.460 ","End":"04:41.294","Text":"Altogether, this answer is equal to 3."},{"Start":"04:41.294 ","End":"04:43.750","Text":"That\u0027s our limit."}],"ID":1585},{"Watched":false,"Name":"Exercise 5","Duration":"2m 26s","ChapterTopicVideoID":1574,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.190","Text":"In this exercise, we have to find the limit as x goes to 0 of x times sine of 1 over x."},{"Start":"00:08.190 ","End":"00:14.350","Text":"We can\u0027t substitute x equals 0 because 1 over 0 is not defined."},{"Start":"00:14.990 ","End":"00:17.670","Text":"We can use some tricks."},{"Start":"00:17.670 ","End":"00:22.665","Text":"The main one we\u0027ve been using lately is this formula theorem."},{"Start":"00:22.665 ","End":"00:25.830","Text":"That if we have the product of two functions,"},{"Start":"00:25.830 ","End":"00:27.540","Text":"f of x times g of x,"},{"Start":"00:27.540 ","End":"00:30.480","Text":"where 1 is bounded and the other tends to 0,"},{"Start":"00:30.480 ","End":"00:33.555","Text":"and the product also tends to 0."},{"Start":"00:33.555 ","End":"00:38.000","Text":"Well, here we definitely have the product of two things."},{"Start":"00:38.000 ","End":"00:42.510","Text":"What if we took f of x?"},{"Start":"00:42.760 ","End":"00:45.380","Text":"Now, which way round am I going to do it?"},{"Start":"00:45.380 ","End":"00:47.210","Text":"Well, x is the one that goes to 0,"},{"Start":"00:47.210 ","End":"00:50.660","Text":"so that will be the g. Of course the letters don\u0027t matter,"},{"Start":"00:50.660 ","End":"00:52.220","Text":"but I want to keep it in order."},{"Start":"00:52.220 ","End":"01:01.505","Text":"F of x will equal the sine of 1 over x,"},{"Start":"01:01.505 ","End":"01:07.850","Text":"and I\u0027ll let g of x be what\u0027s left is just x."},{"Start":"01:07.850 ","End":"01:13.820","Text":"Now, sine of 1 over x is bounded."},{"Start":"01:16.570 ","End":"01:21.560","Text":"It\u0027s defined with x goes to 0 because x is not 0."},{"Start":"01:21.560 ","End":"01:26.525","Text":"The sine of whatever 1 over x comes out to be whatever quantity it is,"},{"Start":"01:26.525 ","End":"01:36.995","Text":"the sine of something is bounded between minus 1 and 1 and it\u0027s in-between those."},{"Start":"01:36.995 ","End":"01:41.345","Text":"Now, as for x, x goes to 0."},{"Start":"01:41.345 ","End":"01:44.750","Text":"When x goes to a limit,"},{"Start":"01:44.750 ","End":"01:47.420","Text":"which in this case is 0 in all these cases."},{"Start":"01:47.420 ","End":"01:50.490","Text":"This goes to 0,"},{"Start":"01:53.740 ","End":"01:57.840","Text":"so f of x, g of x."},{"Start":"01:59.680 ","End":"02:03.950","Text":"The limit as x goes to"},{"Start":"02:03.950 ","End":"02:12.020","Text":"0 of x times sine of 1 over x,"},{"Start":"02:12.020 ","End":"02:16.400","Text":"well, this is our f of x and this is our g of x."},{"Start":"02:16.400 ","End":"02:18.665","Text":"By the rule says it goes to 0,"},{"Start":"02:18.665 ","End":"02:22.130","Text":"so this thing has to also be 0."},{"Start":"02:22.130 ","End":"02:25.770","Text":"That\u0027s the answer. We\u0027re done."}],"ID":1586},{"Watched":false,"Name":"Exercise 6","Duration":"2m 45s","ChapterTopicVideoID":1575,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.840","Text":"In this exercise, we have to find the limit of x as x goes to 0 of this expression."},{"Start":"00:06.840 ","End":"00:12.690","Text":"Note that it\u0027s defined as long as x is not equal to 0."},{"Start":"00:12.690 ","End":"00:16.110","Text":"But we can\u0027t find the limit by substitution because"},{"Start":"00:16.110 ","End":"00:20.310","Text":"the natural log of x is not defined when x is 0,"},{"Start":"00:20.310 ","End":"00:23.790","Text":"it goes to minus infinity from the right."},{"Start":"00:23.790 ","End":"00:26.460","Text":"Anyway, so we can\u0027t substitute."},{"Start":"00:26.460 ","End":"00:29.985","Text":"We\u0027re going to have to use another method and a trick."},{"Start":"00:29.985 ","End":"00:33.915","Text":"We have this formula or rule,"},{"Start":"00:33.915 ","End":"00:35.730","Text":"which comes from the sandwich theorem,"},{"Start":"00:35.730 ","End":"00:38.640","Text":"that if we have 2 functions, one of them f,"},{"Start":"00:38.640 ","End":"00:40.290","Text":"which is bounded and the other one,"},{"Start":"00:40.290 ","End":"00:42.075","Text":"g, which tends to 0,"},{"Start":"00:42.075 ","End":"00:45.470","Text":"and if we multiply them together to get a new function,"},{"Start":"00:45.470 ","End":"00:49.240","Text":"f times g, that it also tends to 0."},{"Start":"00:49.240 ","End":"00:51.480","Text":"Let\u0027s see what we\u0027ll do in our case."},{"Start":"00:51.480 ","End":"00:54.740","Text":"We want to write our function"},{"Start":"00:54.740 ","End":"00:58.970","Text":"as something which is bounded times something which goes to 0."},{"Start":"00:58.970 ","End":"01:03.240","Text":"Clearly, the bounded one is going to be"},{"Start":"01:03.240 ","End":"01:08.240","Text":"the cosine one, because cosine is always between minus 1 and 1."},{"Start":"01:08.240 ","End":"01:13.845","Text":"We\u0027ll take f of x as equaling cosine of"},{"Start":"01:13.845 ","End":"01:19.560","Text":"natural log of x squared."},{"Start":"01:19.560 ","End":"01:22.350","Text":"I think it means the natural log of x all squared,"},{"Start":"01:22.350 ","End":"01:25.809","Text":"but it\u0027s not important."},{"Start":"01:26.660 ","End":"01:30.210","Text":"The one that goes to 0 will be x squared."},{"Start":"01:30.210 ","End":"01:36.615","Text":"We\u0027ll take our g of x to equal x squared."},{"Start":"01:36.615 ","End":"01:40.955","Text":"Now, this one fulfills what f is supposed to do,"},{"Start":"01:40.955 ","End":"01:43.530","Text":"that this one is bounded."},{"Start":"01:43.960 ","End":"01:48.335","Text":"As we said, cosine is between minus 1 and 1,"},{"Start":"01:48.335 ","End":"01:51.320","Text":"and x squared,"},{"Start":"01:51.320 ","End":"01:53.420","Text":"obviously when x goes to 0,"},{"Start":"01:53.420 ","End":"01:55.070","Text":"x squared goes to 0,"},{"Start":"01:55.070 ","End":"01:59.809","Text":"so this one goes to 0."},{"Start":"01:59.809 ","End":"02:02.285","Text":"That means that when we multiply them,"},{"Start":"02:02.285 ","End":"02:11.540","Text":"that the limit of x goes to the same as what it was before of f of x times g of x,"},{"Start":"02:11.540 ","End":"02:21.030","Text":"or we\u0027ll just write it as x squared cosine natural log of x squared,"},{"Start":"02:21.030 ","End":"02:30.235","Text":"where this up to here is f and up to here is g of x."},{"Start":"02:30.235 ","End":"02:34.670","Text":"Because of this law rule,"},{"Start":"02:34.670 ","End":"02:36.800","Text":"this product tends to 0,"},{"Start":"02:36.800 ","End":"02:45.749","Text":"so that means that the limit is 0 as x goes to 0 and that\u0027s it."}],"ID":1587},{"Watched":false,"Name":"Exercise 7","Duration":"6m 44s","ChapterTopicVideoID":1576,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.900 ","End":"00:08.130","Text":"infinity of this somewhat complicated expression."},{"Start":"00:08.130 ","End":"00:13.795","Text":"The first thing we would want to do is just try substituting x equals infinity."},{"Start":"00:13.795 ","End":"00:19.220","Text":"Here we get infinity,"},{"Start":"00:19.220 ","End":"00:25.980","Text":"the arctangent is always bounded."},{"Start":"00:25.980 ","End":"00:28.380","Text":"On the numerator we\u0027re going to get infinity,"},{"Start":"00:28.380 ","End":"00:30.690","Text":"and the bottom we are also going to get"},{"Start":"00:30.690 ","End":"00:35.610","Text":"infinity because this is infinity and this is bounded."},{"Start":"00:35.740 ","End":"00:42.800","Text":"So infinity over infinity is not going to help us very much."},{"Start":"00:42.800 ","End":"00:45.680","Text":"We\u0027re going to use some trick."},{"Start":"00:45.680 ","End":"00:53.250","Text":"Here\u0027s where I\u0027ve written down this law or corollary to the Sandwich Theorem."},{"Start":"00:53.250 ","End":"00:56.900","Text":"That is if we have 2 functions, f and g,"},{"Start":"00:56.900 ","End":"00:59.630","Text":"where 1\u0027s bounded and the other 1 tends to 0,"},{"Start":"00:59.630 ","End":"01:02.210","Text":"then their product also tends to 0."},{"Start":"01:02.210 ","End":"01:05.380","Text":"Now it\u0027s not quite clear how to use that here,"},{"Start":"01:05.380 ","End":"01:09.860","Text":"but you\u0027ll soon see if we start doing what we almost always do,"},{"Start":"01:09.860 ","End":"01:12.635","Text":"which is take out the highest power of x,"},{"Start":"01:12.635 ","End":"01:16.565","Text":"we\u0027re actually going to turn out using this law twice."},{"Start":"01:16.565 ","End":"01:22.930","Text":"Let\u0027s first of all do the algebra and say that this equals"},{"Start":"01:22.930 ","End":"01:28.350","Text":"the limit as x goes"},{"Start":"01:28.350 ","End":"01:34.290","Text":"to infinity of,"},{"Start":"01:34.290 ","End":"01:40.950","Text":"take the x out of the top and take the x out of the bottom."},{"Start":"01:40.950 ","End":"01:47.130","Text":"Here after we take x out we get 3 plus 1 over"},{"Start":"01:47.130 ","End":"01:55.995","Text":"x arctangent of 2x minus 3."},{"Start":"01:55.995 ","End":"02:03.350","Text":"Here we\u0027ll get x times 4 plus 1 over x because we took out"},{"Start":"02:03.350 ","End":"02:13.310","Text":"the x times arctangent x minus natural log of x."},{"Start":"02:13.310 ","End":"02:15.874","Text":"Let\u0027s make this a bit longer."},{"Start":"02:15.874 ","End":"02:18.680","Text":"The x\u0027s cancels is not 0,"},{"Start":"02:18.680 ","End":"02:20.645","Text":"it goes to infinity."},{"Start":"02:20.645 ","End":"02:23.135","Text":"Here we have a couple of numbers."},{"Start":"02:23.135 ","End":"02:29.005","Text":"What we have to find is this expression, and this expression."},{"Start":"02:29.005 ","End":"02:31.105","Text":"In each of these, we\u0027ll use this law."},{"Start":"02:31.105 ","End":"02:33.680","Text":"Let\u0027s write the first 1."},{"Start":"02:37.100 ","End":"02:44.270","Text":"There\u0027s brackets missing here and here, sorry."},{"Start":"02:44.270 ","End":"02:48.110","Text":"This 1, let\u0027s call it say,"},{"Start":"02:48.110 ","End":"02:57.510","Text":"asterisk will do at the side including this,"},{"Start":"02:57.510 ","End":"03:00.090","Text":"we\u0027ll call that double asterisk."},{"Start":"03:00.090 ","End":"03:03.840","Text":"That will be asterisk, asterisk."},{"Start":"03:03.840 ","End":"03:09.450","Text":"Let\u0027s do these sub exercises at the side."},{"Start":"03:09.450 ","End":"03:13.350","Text":"First of all we\u0027ll do the 1 called asterisk."},{"Start":"03:13.350 ","End":"03:20.835","Text":"What we have is we need to find the limit as x goes to infinity"},{"Start":"03:20.835 ","End":"03:29.530","Text":"of 1 over x times arctangent of 2x minus 3."},{"Start":"03:29.530 ","End":"03:32.150","Text":"Now, we\u0027ve done several of these,"},{"Start":"03:32.150 ","End":"03:35.195","Text":"so you\u0027re probably familiar with the technique."},{"Start":"03:35.195 ","End":"03:38.660","Text":"We split this up into 2 pieces."},{"Start":"03:38.660 ","End":"03:44.590","Text":"1 of them is f, 1 of them is g. In this case, the f of x"},{"Start":"03:55.040 ","End":"03:58.815","Text":"is the arctangent of 2x minus 3."},{"Start":"03:58.815 ","End":"04:04.995","Text":"This is certainly bounded because arctangent is bounded in the range"},{"Start":"04:04.995 ","End":"04:12.565","Text":"minus Pi over 2, Pi over 2."},{"Start":"04:12.565 ","End":"04:18.310","Text":"It can\u0027t go below minus Pi over 2 and it can\u0027t go above Pi over 2."},{"Start":"04:18.310 ","End":"04:20.305","Text":"That\u0027s bounded."},{"Start":"04:20.305 ","End":"04:26.950","Text":"The g of x, which is 1 over x,"},{"Start":"04:26.950 ","End":"04:29.440","Text":"certainly tends to 0."},{"Start":"04:29.440 ","End":"04:32.660","Text":"This 1 is bounded."},{"Start":"04:34.430 ","End":"04:38.995","Text":"This 1 goes to 0."},{"Start":"04:38.995 ","End":"04:42.385","Text":"When x goes to infinity,"},{"Start":"04:42.385 ","End":"04:44.650","Text":"this goes to 0."},{"Start":"04:44.650 ","End":"04:48.430","Text":"We have the case of bounded times 0."},{"Start":"04:48.430 ","End":"04:50.900","Text":"The product also goes to 0."},{"Start":"04:50.900 ","End":"04:53.060","Text":"So f of x, g of x,"},{"Start":"04:53.060 ","End":"04:56.615","Text":"which is 1 of x,"},{"Start":"04:56.615 ","End":"05:01.490","Text":"times limit 1 over x,"},{"Start":"05:01.490 ","End":"05:06.860","Text":"which is that g part times f of x,"},{"Start":"05:06.860 ","End":"05:13.970","Text":"which is the arctangent of 2x minus 3."},{"Start":"05:13.970 ","End":"05:21.980","Text":"As x goes to infinity is also going to equal 0,"},{"Start":"05:21.980 ","End":"05:28.590","Text":"and that\u0027s our other 1 we call the asterisk."},{"Start":"05:32.570 ","End":"05:35.865","Text":"In almost exactly the same way,"},{"Start":"05:35.865 ","End":"05:40.465","Text":"in the denominator, we also have something bounded,"},{"Start":"05:40.465 ","End":"05:43.090","Text":"which is the arctangent,"},{"Start":"05:43.090 ","End":"05:45.760","Text":"at just something else in here but it\u0027s still bounded in"},{"Start":"05:45.760 ","End":"05:49.370","Text":"the same range in the 1 over x still goes to 0."},{"Start":"05:50.270 ","End":"05:53.830","Text":"It\u0027s going to be exactly the same but the limit of"},{"Start":"05:53.830 ","End":"05:58.600","Text":"the second thing is also going to be what I call the 2"},{"Start":"05:58.600 ","End":"06:05.420","Text":"stars limit x goes"},{"Start":"06:05.420 ","End":"06:11.095","Text":"to infinity of 1 over x arctangent of whatever."},{"Start":"06:11.095 ","End":"06:18.140","Text":"In this case x minus natural log of x is also going to be equal to 0,"},{"Start":"06:18.140 ","End":"06:21.065","Text":"and that\u0027s our double asterisk."},{"Start":"06:21.065 ","End":"06:24.230","Text":"If I plug both of these back in here,"},{"Start":"06:24.230 ","End":"06:29.364","Text":"what I\u0027m going to get is just equal to 3"},{"Start":"06:29.364 ","End":"06:37.835","Text":"plus 0 over 4 plus 0,"},{"Start":"06:37.835 ","End":"06:40.680","Text":"which is 3/4."},{"Start":"06:40.900 ","End":"06:44.070","Text":"That\u0027s the answer."}],"ID":1588},{"Watched":false,"Name":"Exercise 8","Duration":"6m 5s","ChapterTopicVideoID":1578,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:04.020 ","End":"00:08.295","Text":"infinity of, taught me to pronounce this,"},{"Start":"00:08.295 ","End":"00:11.190","Text":"but this symbol, if it was 20,"},{"Start":"00:11.190 ","End":"00:12.750","Text":"you\u0027d say the 20th root."},{"Start":"00:12.750 ","End":"00:16.455","Text":"If it\u0027s x, y, It\u0027s the xth root of a."},{"Start":"00:16.455 ","End":"00:21.115","Text":"Any event is defined as a to the power of 1 over x."},{"Start":"00:21.115 ","End":"00:23.675","Text":"Excuse me if I mispronounced it."},{"Start":"00:23.675 ","End":"00:26.130","Text":"The limit of this expression,"},{"Start":"00:26.130 ","End":"00:29.350","Text":"the xth root of, so on."},{"Start":"00:29.660 ","End":"00:32.940","Text":"If we just straight forward substitute,"},{"Start":"00:32.940 ","End":"00:35.669","Text":"we get 2 to the infinity is infinity."},{"Start":"00:35.669 ","End":"00:42.665","Text":"Altogether infinity under here and this means that we take to the power of 1 over x."},{"Start":"00:42.665 ","End":"00:46.700","Text":"Essentially, what we would get would be,"},{"Start":"00:46.700 ","End":"00:49.715","Text":"let us write it down here."},{"Start":"00:49.715 ","End":"00:57.410","Text":"We would get infinity to the power of 1 over infinity,"},{"Start":"00:57.410 ","End":"01:00.215","Text":"or if you like infinity to the power of 0."},{"Start":"01:00.215 ","End":"01:03.890","Text":"But this is 1 of these undefined things."},{"Start":"01:03.890 ","End":"01:05.735","Text":"We don\u0027t know what it is,"},{"Start":"01:05.735 ","End":"01:08.825","Text":"so we\u0027re going to have to use some other method."},{"Start":"01:08.825 ","End":"01:12.394","Text":"I suggest take something outside the brackets,"},{"Start":"01:12.394 ","End":"01:14.915","Text":"let\u0027s say 4 to the x."},{"Start":"01:14.915 ","End":"01:23.525","Text":"This equals the limit of"},{"Start":"01:23.525 ","End":"01:28.490","Text":"the xth root of 4 to"},{"Start":"01:28.490 ","End":"01:37.260","Text":"the x times 2 to the x over 4 to the x,"},{"Start":"01:37.260 ","End":"01:42.370","Text":"plus 3 to the x over 4 to the x,"},{"Start":"01:43.250 ","End":"01:48.985","Text":"plus 4 to the x over 4 to the x."},{"Start":"01:48.985 ","End":"01:53.720","Text":"Now let\u0027s remember some of our laws of exponents."},{"Start":"01:53.720 ","End":"01:58.490","Text":"We have here going to do several things."},{"Start":"01:58.490 ","End":"02:00.020","Text":"We\u0027re going to split it up here."},{"Start":"02:00.020 ","End":"02:02.420","Text":"The product of this times this so we take"},{"Start":"02:02.420 ","End":"02:06.350","Text":"the root of each of them without brackets there,"},{"Start":"02:06.350 ","End":"02:12.620","Text":"and we\u0027re also going to use the power of a power,"},{"Start":"02:12.620 ","End":"02:15.575","Text":"something to the power of a power."},{"Start":"02:15.575 ","End":"02:18.995","Text":"Well, might as well say it, what I\u0027m going to use,"},{"Start":"02:18.995 ","End":"02:22.925","Text":"I\u0027m going to use, for example,"},{"Start":"02:22.925 ","End":"02:26.225","Text":"the square root or any root,"},{"Start":"02:26.225 ","End":"02:32.675","Text":"the root whatever of ab equals the root,"},{"Start":"02:32.675 ","End":"02:38.890","Text":"of whatever of a times the root, whatever b."},{"Start":"02:38.890 ","End":"02:39.950","Text":"Whether it\u0027s cube root,"},{"Start":"02:39.950 ","End":"02:43.080","Text":"square root, 20th root, whatever."},{"Start":"02:43.780 ","End":"02:54.440","Text":"The other thing is that a to the power of b to the power of c is a to the power of bc."},{"Start":"02:54.440 ","End":"03:05.690","Text":"For example, 4 to the x to the 1 over x will just be going to use it as in 4 to the x,"},{"Start":"03:05.690 ","End":"03:08.555","Text":"to the 1 over x according to this rule,"},{"Start":"03:08.555 ","End":"03:10.715","Text":"would be 4 to the power of 1,"},{"Start":"03:10.715 ","End":"03:12.865","Text":"which would just equal."},{"Start":"03:12.865 ","End":"03:20.884","Text":"Here\u0027s an example and this is just by way of example,"},{"Start":"03:20.884 ","End":"03:25.265","Text":"is equal to 4 to the 1 which equals 4."},{"Start":"03:25.265 ","End":"03:34.985","Text":"The other thing we\u0027re going to use is that a fraction that say a to the b"},{"Start":"03:34.985 ","End":"03:44.450","Text":"over a to the c. That\u0027s not what I meant."},{"Start":"03:44.450 ","End":"03:47.240","Text":"Excuse me. What I\u0027m doing is looking at"},{"Start":"03:47.240 ","End":"03:54.575","Text":"these fractions and so I feel that the formula I need is that"},{"Start":"03:54.575 ","End":"04:03.350","Text":"a to the c over b to the c is a over b to"},{"Start":"04:03.350 ","End":"04:12.850","Text":"the power of c. With all these rules for exponents."},{"Start":"04:12.890 ","End":"04:16.950","Text":"The first is to do the split and then 4 to the x,"},{"Start":"04:16.950 ","End":"04:20.085","Text":"to the 1 over x is just 4."},{"Start":"04:20.085 ","End":"04:28.010","Text":"Next, we\u0027re going to take it in a bracket and then here we have 2 over"},{"Start":"04:28.010 ","End":"04:37.090","Text":"4 to the power of x plus 3 over 4 to the power of x,"},{"Start":"04:37.090 ","End":"04:43.680","Text":"plus 4 over 4, which is 1 and 1 to the anything is 1 plus 1."},{"Start":"04:43.680 ","End":"04:47.975","Text":"We\u0027re getting closer and this equals,"},{"Start":"04:47.975 ","End":"04:49.910","Text":"well, 2 over 4,"},{"Start":"04:49.910 ","End":"04:52.625","Text":"which is a 1/2 is less than 1."},{"Start":"04:52.625 ","End":"05:00.140","Text":"When you take something that\u0027s less than 1 to increasingly higher and higher powers,"},{"Start":"05:00.140 ","End":"05:03.235","Text":"it goes increasingly closer to 0."},{"Start":"05:03.235 ","End":"05:06.595","Text":"This is 0."},{"Start":"05:06.595 ","End":"05:10.460","Text":"I\u0027ll just write it down in small, this goes to 0."},{"Start":"05:10.460 ","End":"05:15.155","Text":"This also 3/4 is less than 1, so higher powers,"},{"Start":"05:15.155 ","End":"05:17.120","Text":"try it to the power of a 100,"},{"Start":"05:17.120 ","End":"05:19.025","Text":"you get very close to 0,"},{"Start":"05:19.025 ","End":"05:21.140","Text":"and 1 stays 1."},{"Start":"05:21.140 ","End":"05:25.230","Text":"If we do all this arithmetic,"},{"Start":"05:28.460 ","End":"05:31.430","Text":"and we took the xth root here."},{"Start":"05:31.430 ","End":"05:36.420","Text":"We also need to the power of 1 over x, forgive me."},{"Start":"05:37.160 ","End":"05:41.110","Text":"1 over x goes to,"},{"Start":"05:41.270 ","End":"05:44.624","Text":"this thing goes to 0."},{"Start":"05:44.624 ","End":"05:48.260","Text":"Altogether, what we\u0027re going to get is 4 times"},{"Start":"05:48.260 ","End":"05:51.950","Text":"0 plus 0 plus 1 is 1 to the power of 0 is still 1."},{"Start":"05:51.950 ","End":"05:56.795","Text":"4 times 1 is 4 and that\u0027s the answer."},{"Start":"05:56.795 ","End":"05:59.195","Text":"Maybe took too many shortcuts here."},{"Start":"05:59.195 ","End":"06:04.800","Text":"I don\u0027t know. We\u0027re done."}],"ID":1590},{"Watched":false,"Name":"Exercise 8 - Alternate","Duration":"3m 3s","ChapterTopicVideoID":1577,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"There is another way to solve this exercise by means of"},{"Start":"00:03.120 ","End":"00:06.360","Text":"what is colloquially called the sandwich theorem,"},{"Start":"00:06.360 ","End":"00:08.160","Text":"and that\u0027s a name worth remembering,"},{"Start":"00:08.160 ","End":"00:12.750","Text":"and I\u0027ll show you with this exercise what the sandwich theorem is all about."},{"Start":"00:12.750 ","End":"00:16.320","Text":"The first idea is to take the function and find"},{"Start":"00:16.320 ","End":"00:20.040","Text":"something more convenient that\u0027s smaller than it."},{"Start":"00:20.040 ","End":"00:21.675","Text":"On the other hand, another thing,"},{"Start":"00:21.675 ","End":"00:23.580","Text":"another function that\u0027s larger than it,"},{"Start":"00:23.580 ","End":"00:26.600","Text":"but we know the limits of both of these uninfected they\u0027re the same."},{"Start":"00:26.600 ","End":"00:27.900","Text":"For this sounds confusing,"},{"Start":"00:27.900 ","End":"00:29.640","Text":"it\u0027ll clear up in the exercise."},{"Start":"00:29.640 ","End":"00:33.000","Text":"What I\u0027m saying here is that our original function,"},{"Start":"00:33.000 ","End":"00:34.930","Text":"which is 2^x,"},{"Start":"00:34.930 ","End":"00:38.250","Text":"plus 3^x, plus 4^x,"},{"Start":"00:38.250 ","End":"00:40.215","Text":"to the power of 1/x."},{"Start":"00:40.215 ","End":"00:44.149","Text":"I\u0027m going to make this smaller than something and also bigger than something."},{"Start":"00:44.149 ","End":"00:49.745","Text":"Now over here, I\u0027ll just drop out these 2 terms and get just 4^x, to the 1/x,"},{"Start":"00:49.745 ","End":"00:53.330","Text":"and on the other hand, if I swell all these,"},{"Start":"00:53.330 ","End":"00:54.560","Text":"this 2 to a 4,"},{"Start":"00:54.560 ","End":"00:55.925","Text":"and the 3 to a 4,"},{"Start":"00:55.925 ","End":"00:58.115","Text":"I\u0027ll get 3 times 4 to the x."},{"Start":"00:58.115 ","End":"01:01.400","Text":"Here we have 3 times 4^x,"},{"Start":"01:01.400 ","End":"01:03.280","Text":"to the power of 1/x,"},{"Start":"01:03.280 ","End":"01:05.255","Text":"and doing a bit of algebra,"},{"Start":"01:05.255 ","End":"01:07.240","Text":"this thing here is just 4,"},{"Start":"01:07.240 ","End":"01:11.575","Text":"so 4 is less than our expression, 2^x,"},{"Start":"01:11.575 ","End":"01:15.230","Text":"plus 3^x, plus 4^x,"},{"Start":"01:15.230 ","End":"01:17.550","Text":"all this to the power of 1/x."},{"Start":"01:17.550 ","End":"01:18.870","Text":"On the other hand,"},{"Start":"01:18.870 ","End":"01:23.750","Text":"here we\u0027ll get 3^1/x times 4."},{"Start":"01:23.750 ","End":"01:27.350","Text":"If I put the word limit in front of each of these,"},{"Start":"01:27.350 ","End":"01:31.550","Text":"then the limit as x goes to infinity,"},{"Start":"01:31.550 ","End":"01:33.260","Text":"well, I\u0027m just 4,"},{"Start":"01:33.260 ","End":"01:39.650","Text":"and here we have the limit as x goes to infinity of 2^x,"},{"Start":"01:39.650 ","End":"01:43.160","Text":"plus 3^x, plus 4^x,"},{"Start":"01:43.160 ","End":"01:45.050","Text":"all this to the 1/x,"},{"Start":"01:45.050 ","End":"01:46.340","Text":"and this is what we\u0027re looking for."},{"Start":"01:46.340 ","End":"01:51.995","Text":"Here, we have 3^1/x, times 4."},{"Start":"01:51.995 ","End":"01:55.920","Text":"I forgot to put the limit here, hang on."},{"Start":"01:55.920 ","End":"01:57.980","Text":"There we go. What I was trying to say,"},{"Start":"01:57.980 ","End":"02:00.055","Text":"I didn\u0027t put any sign between these,"},{"Start":"02:00.055 ","End":"02:02.805","Text":"because although here we have a strict inequality,"},{"Start":"02:02.805 ","End":"02:06.495","Text":"here we only get a non-strict inequality,"},{"Start":"02:06.495 ","End":"02:09.230","Text":"in other words, it could be less than or equal to."},{"Start":"02:09.230 ","End":"02:14.315","Text":"What we get here is that the limit of a constant is just that constant,"},{"Start":"02:14.315 ","End":"02:19.760","Text":"so here we have 4 is less than or equal to the limit that we want,"},{"Start":"02:19.760 ","End":"02:21.815","Text":"is less than or even."},{"Start":"02:21.815 ","End":"02:25.625","Text":"Copy it in the original form, change x,"},{"Start":"02:25.625 ","End":"02:29.295","Text":"x goes to infinity of whatever we had,"},{"Start":"02:29.295 ","End":"02:32.955","Text":"2^x plus 3^x plus 4^x,"},{"Start":"02:32.955 ","End":"02:34.310","Text":"and this limit here,"},{"Start":"02:34.310 ","End":"02:36.575","Text":"4 is just a constant so it comes out."},{"Start":"02:36.575 ","End":"02:38.435","Text":"Now when x goes to infinity,"},{"Start":"02:38.435 ","End":"02:39.950","Text":"1 over infinity is 0,"},{"Start":"02:39.950 ","End":"02:41.480","Text":"3^0 is 1,"},{"Start":"02:41.480 ","End":"02:44.525","Text":"so this comes out to be just also equal to 4."},{"Start":"02:44.525 ","End":"02:47.750","Text":"How can something be both bigger or equal to 4 and"},{"Start":"02:47.750 ","End":"02:51.320","Text":"smaller or equal to 4 unless itself it\u0027s 4?"},{"Start":"02:51.320 ","End":"02:57.110","Text":"This limit is simply equal to 4 and it\u0027s forced to be that way by the sandwich theorem,"},{"Start":"02:57.110 ","End":"03:00.300","Text":"and again, this is our solution and it\u0027s encouraging,"},{"Start":"03:00.300 ","End":"03:03.540","Text":"and that\u0027s what we got before with the other method."}],"ID":1589},{"Watched":false,"Name":"Exercise 9","Duration":"4m 55s","ChapterTopicVideoID":1579,"CourseChapterTopicPlaylistID":169,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.690 ","End":"00:09.374","Text":"infinity of this expression 1 over x times."},{"Start":"00:09.374 ","End":"00:10.860","Text":"Now what is this?"},{"Start":"00:10.860 ","End":"00:13.980","Text":"It\u0027s x inside the square brackets. What does that mean?"},{"Start":"00:13.980 ","End":"00:18.210","Text":"Well, this is the notation we use for something sometimes called"},{"Start":"00:18.210 ","End":"00:23.035","Text":"the greatest integer function and sometimes the floor function."},{"Start":"00:23.035 ","End":"00:29.075","Text":"What it means is the integer closest to our number but from below."},{"Start":"00:29.075 ","End":"00:35.285","Text":"For example, the greatest integer of 4.3 is 4,"},{"Start":"00:35.285 ","End":"00:38.630","Text":"and even a 4.99 it\u0027s 4,"},{"Start":"00:38.630 ","End":"00:45.960","Text":"but the greatest integer of 5 is just 5 itself, and you get the idea."},{"Start":"00:46.370 ","End":"00:51.620","Text":"1 over x times the greatest integer function of x."},{"Start":"00:51.620 ","End":"00:53.990","Text":"Now when x goes to infinity,"},{"Start":"00:53.990 ","End":"00:56.435","Text":"1 over infinity is 0,"},{"Start":"00:56.435 ","End":"01:02.520","Text":"and this thing jumps,"},{"Start":"01:02.520 ","End":"01:06.105","Text":"the limit is actually infinity."},{"Start":"01:06.105 ","End":"01:11.760","Text":"What we get is a 0 times infinity situation,"},{"Start":"01:11.760 ","End":"01:17.120","Text":"so we don\u0027t know which 1 of those indeterminate, undefined forms."},{"Start":"01:19.560 ","End":"01:25.970","Text":"What we\u0027re going to do is make use of the sandwich theorem to help us here."},{"Start":"01:26.150 ","End":"01:29.745","Text":"What we\u0027re going to is use inequalities,"},{"Start":"01:29.745 ","End":"01:34.385","Text":"we\u0027re going to sandwich this thing between something smaller and something greater,"},{"Start":"01:34.385 ","End":"01:38.450","Text":"and that will force it to have a limit."},{"Start":"01:38.850 ","End":"01:44.290","Text":"Let\u0027s start off by writing something that is fairly obvious,"},{"Start":"01:44.290 ","End":"01:51.350","Text":"and that is that the greatest integer of a number x,"},{"Start":"01:51.450 ","End":"02:00.385","Text":"is got to be less than or equal to x,"},{"Start":"02:00.385 ","End":"02:05.649","Text":"but also greater than"},{"Start":"02:05.649 ","End":"02:13.390","Text":"x minus 1 will never round down by more than 1."},{"Start":"02:13.390 ","End":"02:16.620","Text":"If we have 4.99,"},{"Start":"02:16.620 ","End":"02:18.195","Text":"we can go down to 4,"},{"Start":"02:18.195 ","End":"02:23.040","Text":"but we won\u0027t go down to 3.99 or something."},{"Start":"02:23.040 ","End":"02:27.850","Text":"This inequality holds."},{"Start":"02:28.430 ","End":"02:33.965","Text":"If we divide by x will also hold,"},{"Start":"02:33.965 ","End":"02:43.655","Text":"so what we have is that x minus 1"},{"Start":"02:43.655 ","End":"02:48.450","Text":"over x is less than"},{"Start":"02:50.560 ","End":"02:59.360","Text":"greatest integer of x over x and is less than or equal to x over x,"},{"Start":"02:59.360 ","End":"03:02.310","Text":"which is just equal to 1."},{"Start":"03:02.900 ","End":"03:06.440","Text":"Now of these 3 quantities,"},{"Start":"03:06.440 ","End":"03:11.525","Text":"we can take the limit as x goes to infinity of the first and the last,"},{"Start":"03:11.525 ","End":"03:16.535","Text":"and if they come out the same that will force the middle 1 to go to that same limit."},{"Start":"03:16.535 ","End":"03:25.440","Text":"Limit as x goes to infinity of x minus 1 over x,"},{"Start":"03:25.440 ","End":"03:30.585","Text":"which is just exactly 1 minus 1 over x."},{"Start":"03:30.585 ","End":"03:33.060","Text":"To substitute infinity,"},{"Start":"03:33.060 ","End":"03:36.120","Text":"1 minus 0 that\u0027s equal to 1."},{"Start":"03:36.120 ","End":"03:38.250","Text":"On the other hand,"},{"Start":"03:38.250 ","End":"03:46.205","Text":"the limit as x goes to infinity of 1,"},{"Start":"03:46.205 ","End":"03:47.670","Text":"it\u0027s a constant function,"},{"Start":"03:47.670 ","End":"03:51.650","Text":"still has a limit, is also just 1."},{"Start":"03:51.650 ","End":"03:57.390","Text":"What we have here is what would go in between them."},{"Start":"03:57.390 ","End":"04:02.955","Text":"This would have to be between this and this, and also 1."},{"Start":"04:02.955 ","End":"04:13.260","Text":"Other words, we have that 1 is less than or equal to the limit"},{"Start":"04:13.260 ","End":"04:16.835","Text":"as x goes to infinity"},{"Start":"04:16.835 ","End":"04:25.445","Text":"of absolute value of x over x,"},{"Start":"04:25.445 ","End":"04:29.910","Text":"which is also less than or equal to 1."},{"Start":"04:30.250 ","End":"04:34.880","Text":"What we\u0027ve written here is just the original,"},{"Start":"04:34.880 ","End":"04:37.895","Text":"what was written originally is what we\u0027re looking for,"},{"Start":"04:37.895 ","End":"04:40.130","Text":"just written slightly differently."},{"Start":"04:40.130 ","End":"04:48.990","Text":"This automatically has to equal 1 because it\u0027s trapped between 1 and 1,"},{"Start":"04:50.140 ","End":"04:54.240","Text":"so that is the answer."}],"ID":1591}],"Thumbnail":null,"ID":169},{"Name":"Technique 9 Piecewise Functions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Limit of Piecewise Functions","Duration":"7m 9s","ChapterTopicVideoID":8253,"CourseChapterTopicPlaylistID":170,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"Now we come to technique number 9,"},{"Start":"00:02.820 ","End":"00:07.875","Text":"limits of piecewise defined functions or just piecewise functions."},{"Start":"00:07.875 ","End":"00:10.890","Text":"Let me just start right away with an example."},{"Start":"00:10.890 ","End":"00:13.245","Text":"F of x equals and usually,"},{"Start":"00:13.245 ","End":"00:18.645","Text":"you can expect to see curly braces because piecewise means to find in more than 1 piece,"},{"Start":"00:18.645 ","End":"00:20.594","Text":"in this case 2 pieces."},{"Start":"00:20.594 ","End":"00:27.270","Text":"We\u0027ll define it as x squared for those x which are less than 1,"},{"Start":"00:27.270 ","End":"00:33.265","Text":"and we\u0027ll define f of x equals x for x greater than 1."},{"Start":"00:33.265 ","End":"00:36.515","Text":"Note that it\u0027s not defined at all for x equals 1,"},{"Start":"00:36.515 ","End":"00:39.005","Text":"so there\u0027s going to be a hole in the graph."},{"Start":"00:39.005 ","End":"00:41.560","Text":"Let\u0027s start straight away with the sketch of it,"},{"Start":"00:41.560 ","End":"00:43.200","Text":"and let\u0027s say that this,"},{"Start":"00:43.200 ","End":"00:44.550","Text":"just to give you some scale,"},{"Start":"00:44.550 ","End":"00:46.215","Text":"that this is 0,"},{"Start":"00:46.215 ","End":"00:48.420","Text":"here somewhere is 1,"},{"Start":"00:48.420 ","End":"00:50.890","Text":"here somewhere, minus 1."},{"Start":"00:50.890 ","End":"00:53.000","Text":"I want the point 1 here."},{"Start":"00:53.000 ","End":"00:54.665","Text":"That\u0027s really all I\u0027ll need."},{"Start":"00:54.665 ","End":"00:58.970","Text":"Now the first bit where f of x is x squared for x less than 1,"},{"Start":"00:58.970 ","End":"01:01.805","Text":"so everything up to but not including the 1,"},{"Start":"01:01.805 ","End":"01:05.180","Text":"will be like a parabola, something like this."},{"Start":"01:05.180 ","End":"01:06.620","Text":"But I\u0027ve got to stop."},{"Start":"01:06.620 ","End":"01:09.570","Text":"Once we get to 1 on the other side,"},{"Start":"01:09.570 ","End":"01:10.980","Text":"it\u0027s something like this,"},{"Start":"01:10.980 ","End":"01:13.080","Text":"then it goes through minus 1,"},{"Start":"01:13.080 ","End":"01:14.880","Text":"1, goes through 0,"},{"Start":"01:14.880 ","End":"01:17.240","Text":"0, and if it was to continue,"},{"Start":"01:17.240 ","End":"01:18.680","Text":"it would go through 1,"},{"Start":"01:18.680 ","End":"01:20.230","Text":"1 only it doesn\u0027t,"},{"Start":"01:20.230 ","End":"01:22.075","Text":"there\u0027s a hole there,"},{"Start":"01:22.075 ","End":"01:23.810","Text":"not defined for x equals 1."},{"Start":"01:23.810 ","End":"01:26.750","Text":"For x bigger than 1, f of x equals x is"},{"Start":"01:26.750 ","End":"01:30.110","Text":"a straight line through the origin and going through 1,"},{"Start":"01:30.110 ","End":"01:36.145","Text":"1, so I continue from here like a straight line or it\u0027s supposed to be,"},{"Start":"01:36.145 ","End":"01:37.460","Text":"and this is the graph."},{"Start":"01:37.460 ","End":"01:41.630","Text":"The whole thing is y equals f of x,"},{"Start":"01:41.630 ","End":"01:43.880","Text":"where the limits come into this."},{"Start":"01:43.880 ","End":"01:48.260","Text":"Well, really the only interesting values that wish to try and take the limit,"},{"Start":"01:48.260 ","End":"01:52.460","Text":"the point where it split the seam points or seam lines."},{"Start":"01:52.460 ","End":"01:55.960","Text":"Basically x equals 1 is where something happens."},{"Start":"01:55.960 ","End":"01:58.700","Text":"I\u0027ll just say something about what happens at the other points."},{"Start":"01:58.700 ","End":"02:00.200","Text":"If I just take a regular point,"},{"Start":"02:00.200 ","End":"02:02.660","Text":"for example, if I said,"},{"Start":"02:02.660 ","End":"02:10.315","Text":"what is the limit as x goes to minus 2 of f of x?"},{"Start":"02:10.315 ","End":"02:12.380","Text":"Minus 2 is off the chart,"},{"Start":"02:12.380 ","End":"02:13.880","Text":"but that doesn\u0027t matter."},{"Start":"02:13.880 ","End":"02:16.115","Text":"Minus 2 belongs to this bit."},{"Start":"02:16.115 ","End":"02:17.975","Text":"Minus 2 is less than 1,"},{"Start":"02:17.975 ","End":"02:23.930","Text":"so I would say that this is equal to minus 2 squared, which is equal to 4."},{"Start":"02:23.930 ","End":"02:25.820","Text":"I just do it by substitution."},{"Start":"02:25.820 ","End":"02:27.610","Text":"Nothing interesting there."},{"Start":"02:27.610 ","End":"02:30.890","Text":"If I asked you what is the limit as x goes to,"},{"Start":"02:30.890 ","End":"02:33.035","Text":"let\u0027s say, 2 of f of x?"},{"Start":"02:33.035 ","End":"02:34.485","Text":"I would say, oh yeah,"},{"Start":"02:34.485 ","End":"02:35.790","Text":"2 is bigger than 1,"},{"Start":"02:35.790 ","End":"02:39.815","Text":"so it\u0027s equal to just 2 and nothing interesting there."},{"Start":"02:39.815 ","End":"02:41.780","Text":"Just get it by substitution."},{"Start":"02:41.780 ","End":"02:44.150","Text":"When we talk about limits of piecewise functions,"},{"Start":"02:44.150 ","End":"02:50.090","Text":"the only interesting thing is what happens at these seam line points or whatever."},{"Start":"02:50.090 ","End":"02:56.855","Text":"So what I want to know is what is the limit as x goes to 1 of f of x?"},{"Start":"02:56.855 ","End":"02:58.700","Text":"That\u0027s the question."},{"Start":"02:58.700 ","End":"03:02.360","Text":"Notice that the fact that f is not defined at 1 doesn\u0027t"},{"Start":"03:02.360 ","End":"03:06.020","Text":"bother me because to have a limit as x tends to a point,"},{"Start":"03:06.020 ","End":"03:09.895","Text":"the function doesn\u0027t have to be defined at the point Often it isn\u0027t."},{"Start":"03:09.895 ","End":"03:13.240","Text":"The technique of doing such a thing is,"},{"Start":"03:13.240 ","End":"03:15.175","Text":"so if it\u0027s on the boarder,"},{"Start":"03:15.175 ","End":"03:16.685","Text":"if it\u0027s a seam line,"},{"Start":"03:16.685 ","End":"03:21.480","Text":"we have to do 2 separate things to approach from the right and to approach from the left."},{"Start":"03:21.480 ","End":"03:22.580","Text":"So on the 1 hand,"},{"Start":"03:22.580 ","End":"03:26.850","Text":"we\u0027ll be taking the right-handed limit as we get to 1 from the right."},{"Start":"03:26.850 ","End":"03:29.820","Text":"On the other hand, we\u0027ll be going from the left,"},{"Start":"03:29.820 ","End":"03:34.670","Text":"and on the graph it\u0027s like we\u0027re approaching 1 way and we\u0027re approaching the other way."},{"Start":"03:34.670 ","End":"03:36.320","Text":"Now in this particular case,"},{"Start":"03:36.320 ","End":"03:40.100","Text":"we can see from the picture that we do have a limit."},{"Start":"03:40.100 ","End":"03:45.290","Text":"It looks like the right limit and the left limit are equal and they\u0027re both equal to 1,"},{"Start":"03:45.290 ","End":"03:48.035","Text":"and so in this case we would say the limit is 1."},{"Start":"03:48.035 ","End":"03:53.005","Text":"But in general, what we do is we break it up into 2 sub-problems."},{"Start":"03:53.005 ","End":"04:00.200","Text":"We ask, is there a limit as x goes to 1 from the right of f of x?"},{"Start":"04:00.200 ","End":"04:06.945","Text":"Is there a limit as x goes to 1 from the left of f of x?"},{"Start":"04:06.945 ","End":"04:09.590","Text":"If the answer to both of these is yes,"},{"Start":"04:09.590 ","End":"04:11.270","Text":"the limits exist, but more than that,"},{"Start":"04:11.270 ","End":"04:12.710","Text":"that they are equal,"},{"Start":"04:12.710 ","End":"04:14.525","Text":"they both have to be equal,"},{"Start":"04:14.525 ","End":"04:17.375","Text":"then we would say that that\u0027s the limit for this."},{"Start":"04:17.375 ","End":"04:19.550","Text":"Now, in our case,"},{"Start":"04:19.550 ","End":"04:21.440","Text":"if we went from the right,"},{"Start":"04:21.440 ","End":"04:23.540","Text":"x is bigger than 1,"},{"Start":"04:23.540 ","End":"04:25.400","Text":"and even if it\u0027s very close to 1,"},{"Start":"04:25.400 ","End":"04:26.780","Text":"it\u0027s still bigger than 1,"},{"Start":"04:26.780 ","End":"04:28.855","Text":"and so we use this definition."},{"Start":"04:28.855 ","End":"04:35.780","Text":"This would be the limit as x goes to 1 from the right of x squared,"},{"Start":"04:35.780 ","End":"04:37.100","Text":"and this is equal to,"},{"Start":"04:37.100 ","End":"04:40.699","Text":"by substitution, is equal to 1 squared, which is 1."},{"Start":"04:40.699 ","End":"04:43.415","Text":"Just let me change the left and the right."},{"Start":"04:43.415 ","End":"04:46.370","Text":"From the left we have x squared,"},{"Start":"04:46.370 ","End":"04:49.900","Text":"and from the right we have the limit x,"},{"Start":"04:49.900 ","End":"04:52.535","Text":"and that\u0027s equal to just 1 itself."},{"Start":"04:52.535 ","End":"04:57.485","Text":"Now because this exists and this exists, and more than that,"},{"Start":"04:57.485 ","End":"05:04.730","Text":"they\u0027re both equal, then we can conclude that this limit is also equal to 1."},{"Start":"05:04.730 ","End":"05:08.185","Text":"I can now erase this and write 1."},{"Start":"05:08.185 ","End":"05:13.610","Text":"I\u0027ll erase this example and the next example we\u0027ll take as x"},{"Start":"05:13.610 ","End":"05:19.445","Text":"squared plus 3 when x is bigger than 2,"},{"Start":"05:19.445 ","End":"05:25.250","Text":"and I\u0027ll take it as 2x minus 4 when x is less than 2."},{"Start":"05:25.250 ","End":"05:28.790","Text":"In fact, why don\u0027t I even make this greater or equal to?"},{"Start":"05:28.790 ","End":"05:30.679","Text":"It actually makes no difference."},{"Start":"05:30.679 ","End":"05:35.680","Text":"It doesn\u0027t really matter what happens when x equals 2, and of course,"},{"Start":"05:35.680 ","End":"05:37.535","Text":"my question is going to be,"},{"Start":"05:37.535 ","End":"05:42.150","Text":"what is the limit as x goes to 2 of f of x,"},{"Start":"05:42.150 ","End":"05:43.785","Text":"or does it even exist?"},{"Start":"05:43.785 ","End":"05:45.830","Text":"This time we\u0027ll do it without a sketch."},{"Start":"05:45.830 ","End":"05:48.935","Text":"So what we do is we check 2 different things."},{"Start":"05:48.935 ","End":"05:51.215","Text":"We check limit from the right,"},{"Start":"05:51.215 ","End":"05:56.650","Text":"x goes to 2 from the right of f of x and see what that is,"},{"Start":"05:56.650 ","End":"06:04.665","Text":"and then we\u0027ll see what is limit as x goes to 2 from the left and see what that is,"},{"Start":"06:04.665 ","End":"06:08.070","Text":"and see if we get 2 equal things."},{"Start":"06:08.070 ","End":"06:10.160","Text":"If we\u0027re going from the right,"},{"Start":"06:10.160 ","End":"06:12.660","Text":"then we using this formula,"},{"Start":"06:12.660 ","End":"06:14.115","Text":"when x is bigger than 2,"},{"Start":"06:14.115 ","End":"06:15.480","Text":"I don\u0027t care about the equal 2,"},{"Start":"06:15.480 ","End":"06:16.530","Text":"but if it\u0027s bigger than 2,"},{"Start":"06:16.530 ","End":"06:18.390","Text":"then it\u0027s x squared plus 3,"},{"Start":"06:18.390 ","End":"06:23.090","Text":"so its limit as x goes to 2 from the right of x"},{"Start":"06:23.090 ","End":"06:27.709","Text":"squared plus 3, elementary polynomial function."},{"Start":"06:27.709 ","End":"06:29.480","Text":"Just substitute the 2,"},{"Start":"06:29.480 ","End":"06:31.280","Text":"doesn\u0027t matter from the right or the left."},{"Start":"06:31.280 ","End":"06:37.470","Text":"The limit as x goes to 2 of this thing is 2 squared plus 3, which is 7."},{"Start":"06:37.470 ","End":"06:38.820","Text":"From the other side,"},{"Start":"06:38.820 ","End":"06:41.104","Text":"I just use the other formula."},{"Start":"06:41.104 ","End":"06:46.625","Text":"It\u0027s the limit as x goes to 2 from the left of 2x minus 4."},{"Start":"06:46.625 ","End":"06:50.930","Text":"The left doesn\u0027t really matter because this thing has a limit when x goes to 2,"},{"Start":"06:50.930 ","End":"06:56.395","Text":"and just get it by substitution twice 2 minus 4, which is 0."},{"Start":"06:56.395 ","End":"06:58.225","Text":"I have a limit from the right,"},{"Start":"06:58.225 ","End":"07:00.170","Text":"and I have a limit from the left,"},{"Start":"07:00.170 ","End":"07:03.335","Text":"but they are not equal."},{"Start":"07:03.335 ","End":"07:07.370","Text":"From this, I conclude that the limit does not exist."},{"Start":"07:07.370 ","End":"07:10.560","Text":"That\u0027s all I can say for this."}],"ID":8413},{"Watched":false,"Name":"Exercise 1","Duration":"4m 13s","ChapterTopicVideoID":1580,"CourseChapterTopicPlaylistID":170,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.350 ","End":"00:07.965","Text":"In this exercise, we have to find the limit as x tends to 0 of the function f,"},{"Start":"00:07.965 ","End":"00:11.655","Text":"where f is defined piecewise."},{"Start":"00:11.655 ","End":"00:13.500","Text":"When x is bigger than 0,"},{"Start":"00:13.500 ","End":"00:14.970","Text":"it\u0027s defined one way,"},{"Start":"00:14.970 ","End":"00:17.355","Text":"and when x is less than 0,"},{"Start":"00:17.355 ","End":"00:18.990","Text":"it\u0027s defined a different way."},{"Start":"00:18.990 ","End":"00:23.860","Text":"In fact, it\u0027s not even defined when x actually equals 0."},{"Start":"00:24.440 ","End":"00:28.050","Text":"In the case of this sort,"},{"Start":"00:28.050 ","End":"00:32.490","Text":"we find the limit on the left and the limit on the right."},{"Start":"00:32.490 ","End":"00:42.920","Text":"In other words, we have to find the limit as x goes to 0"},{"Start":"00:42.920 ","End":"00:50.285","Text":"from the right of f of x and see what is that equal to,"},{"Start":"00:50.285 ","End":"01:01.220","Text":"and we also have to find the limit as x goes to 0 from the left of f of x."},{"Start":"01:01.220 ","End":"01:05.870","Text":"Only if they both turn out to be equal,"},{"Start":"01:05.870 ","End":"01:08.525","Text":"then we can say that we really have a limit,"},{"Start":"01:08.525 ","End":"01:09.680","Text":"and that\u0027s what it would be,"},{"Start":"01:09.680 ","End":"01:15.880","Text":"the common value of the limit on the left and the limit on the right."},{"Start":"01:16.400 ","End":"01:19.895","Text":"When x goes to 0 from the right,"},{"Start":"01:19.895 ","End":"01:22.174","Text":"then we\u0027re going to use this formula."},{"Start":"01:22.174 ","End":"01:37.580","Text":"This is equal to limit as x goes to 0 plus sine 4x over x."},{"Start":"01:37.580 ","End":"01:40.340","Text":"These kind of limits are familiar to us,"},{"Start":"01:40.340 ","End":"01:44.870","Text":"so I\u0027ll just keep on doing what we usually do and say that"},{"Start":"01:44.870 ","End":"01:52.790","Text":"this is the limit of sine 4x over 4x,"},{"Start":"01:52.790 ","End":"01:55.800","Text":"and that\u0027s a familiar pattern,"},{"Start":"01:55.800 ","End":"01:59.580","Text":"sine of something over that same something,"},{"Start":"01:59.580 ","End":"02:02.780","Text":"but we change the exercise with this 4 so"},{"Start":"02:02.780 ","End":"02:06.400","Text":"we have to compensate by putting it as times 4,"},{"Start":"02:06.400 ","End":"02:11.055","Text":"and of course, x goes to 0 plus."},{"Start":"02:11.055 ","End":"02:14.560","Text":"This limit, as I say, is familiar."},{"Start":"02:16.400 ","End":"02:19.155","Text":"This limit is equal to 1,"},{"Start":"02:19.155 ","End":"02:21.075","Text":"and 4 is just 4,"},{"Start":"02:21.075 ","End":"02:23.270","Text":"so this is equal to 1 times 4,"},{"Start":"02:23.270 ","End":"02:25.595","Text":"so this is equal to 4."},{"Start":"02:25.595 ","End":"02:31.905","Text":"That\u0027s the limit on the right."},{"Start":"02:31.905 ","End":"02:34.805","Text":"Now let\u0027s see what happens on the left."},{"Start":"02:34.805 ","End":"02:39.455","Text":"On the left, meaning x goes to 0 but is slightly negative,"},{"Start":"02:39.455 ","End":"02:41.750","Text":"so we use this formula."},{"Start":"02:41.750 ","End":"02:44.160","Text":"This is the limit."},{"Start":"02:46.190 ","End":"02:51.150","Text":"x goes to 0 from the left"},{"Start":"02:51.150 ","End":"03:00.575","Text":"of 4 plus e to the power of 1 over x."},{"Start":"03:00.575 ","End":"03:06.895","Text":"Here, we have to substitute 0 minus."},{"Start":"03:06.895 ","End":"03:09.870","Text":"This part is easy, that\u0027s just 4,"},{"Start":"03:09.870 ","End":"03:12.640","Text":"and this part,"},{"Start":"03:24.200 ","End":"03:32.435","Text":"what we get is e to the minus infinity because when x goes to 0 minus,"},{"Start":"03:32.435 ","End":"03:39.920","Text":"then 1 over x goes to minus infinity and e to the minus infinity is equal to 0,"},{"Start":"03:39.920 ","End":"03:43.230","Text":"so this just comes out to be 4."},{"Start":"03:43.550 ","End":"03:47.930","Text":"This is what we have on the limit on the right,"},{"Start":"03:47.930 ","End":"03:51.635","Text":"this is the limit on the left,"},{"Start":"03:51.635 ","End":"03:53.390","Text":"and since they\u0027re both equal,"},{"Start":"03:53.390 ","End":"03:55.655","Text":"then we can say, altogether,"},{"Start":"03:55.655 ","End":"04:01.610","Text":"that the limit as x goes to 0 without"},{"Start":"04:01.610 ","End":"04:08.150","Text":"saying plus or minus of f of x is equal to 4,"},{"Start":"04:08.150 ","End":"04:10.340","Text":"because the right and the left is the same."},{"Start":"04:10.340 ","End":"04:13.290","Text":"We\u0027re done."}],"ID":1592},{"Watched":false,"Name":"Exercise 2","Duration":"4m 54s","ChapterTopicVideoID":1581,"CourseChapterTopicPlaylistID":170,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:05.430","Text":"In this exercise, we have to find the limit as x goes to 1 of f of x,"},{"Start":"00:05.430 ","End":"00:07.755","Text":"where f of x is defined piecewise."},{"Start":"00:07.755 ","End":"00:10.260","Text":"When x is bigger than 1 it\u0027s defined this way."},{"Start":"00:10.260 ","End":"00:12.890","Text":"When x is less than 1 it\u0027s defined this way."},{"Start":"00:12.890 ","End":"00:17.850","Text":"In actual fact, its x is not even defined for x equals 1,"},{"Start":"00:17.850 ","End":"00:19.920","Text":"but that doesn\u0027t bother us."},{"Start":"00:19.920 ","End":"00:21.870","Text":"We can still find its limit."},{"Start":"00:21.870 ","End":"00:28.710","Text":"What we do as usual is to find out the limit on the left and on the right separately."},{"Start":"00:28.710 ","End":"00:31.890","Text":"If they happen to equal,"},{"Start":"00:31.890 ","End":"00:34.095","Text":"then we\u0027ve got a limit."},{"Start":"00:34.095 ","End":"00:37.275","Text":"First of all, let\u0027s try on the right."},{"Start":"00:37.275 ","End":"00:45.899","Text":"What we get is the limit as x goes to 1"},{"Start":"00:45.899 ","End":"00:56.610","Text":"plus of x squared plus x minus 2 over x minus 1."},{"Start":"00:56.900 ","End":"01:00.440","Text":"Here we try substituting x equals 1,"},{"Start":"01:00.440 ","End":"01:02.150","Text":"1 plus 1 minus 2, 0,"},{"Start":"01:02.150 ","End":"01:05.120","Text":"1 minus 1, 0, so we have 0 over 0."},{"Start":"01:05.120 ","End":"01:06.890","Text":"Well, the usual thing is to try and"},{"Start":"01:06.890 ","End":"01:12.450","Text":"factorize the numerator and hopefully something will cancel."},{"Start":"01:12.700 ","End":"01:17.750","Text":"Now, I\u0027ll just write this at the side."},{"Start":"01:17.750 ","End":"01:20.720","Text":"We\u0027re not going to actually solve the quadratic equation."},{"Start":"01:20.720 ","End":"01:29.245","Text":"I\u0027ll just say that if we tried solving x squared plus x minus 2,"},{"Start":"01:29.245 ","End":"01:32.230","Text":"just the side here."},{"Start":"01:32.230 ","End":"01:35.225","Text":"If we let this equal 0,"},{"Start":"01:35.225 ","End":"01:37.400","Text":"then we get 2 solutions,"},{"Start":"01:37.400 ","End":"01:44.325","Text":"either x equals 1 or x equals minus 2,"},{"Start":"01:44.325 ","End":"01:47.600","Text":"which means that when we factorize this,"},{"Start":"01:47.600 ","End":"01:50.525","Text":"we\u0027re going to get x minus this, x minus that,"},{"Start":"01:50.525 ","End":"01:57.410","Text":"which I mean is x minus"},{"Start":"01:57.410 ","End":"02:07.425","Text":"1 times x plus 2 over x minus 1."},{"Start":"02:07.425 ","End":"02:12.100","Text":"Of course, x goes to 1 from the right."},{"Start":"02:14.810 ","End":"02:18.300","Text":"Good. We have something that cancels."},{"Start":"02:18.300 ","End":"02:21.240","Text":"X minus 1 cancels with x minus 1."},{"Start":"02:21.240 ","End":"02:24.340","Text":"Now, if we let x go to 1,"},{"Start":"02:24.340 ","End":"02:26.240","Text":"then we just get 3."},{"Start":"02:26.240 ","End":"02:28.760","Text":"The answer is equal to 3."},{"Start":"02:28.760 ","End":"02:31.790","Text":"That\u0027s our limit on the right."},{"Start":"02:31.790 ","End":"02:34.475","Text":"Now, let\u0027s try the other 1 on the left."},{"Start":"02:34.475 ","End":"02:41.615","Text":"Here we have limit x goes to 1 from the left."},{"Start":"02:41.615 ","End":"02:43.745","Text":"When it\u0027s less than 1,"},{"Start":"02:43.745 ","End":"02:47.690","Text":"this is the expression."},{"Start":"02:47.690 ","End":"02:58.830","Text":"We have x minus 1 over the square root of x minus 1."},{"Start":"02:59.330 ","End":"03:02.840","Text":"The obvious thing to do here is to"},{"Start":"03:02.840 ","End":"03:07.310","Text":"multiply by the conjugate because we have a square root here."},{"Start":"03:07.310 ","End":"03:08.915","Text":"Let\u0027s just try."},{"Start":"03:08.915 ","End":"03:12.710","Text":"Don\u0027t have to rewrite it, I\u0027ll just continue here by"},{"Start":"03:12.710 ","End":"03:17.660","Text":"multiplying top and bottom by the conjugate of the denominator."},{"Start":"03:17.660 ","End":"03:21.460","Text":"I\u0027ll multiply by something over itself."},{"Start":"03:21.460 ","End":"03:25.020","Text":"Square root of x plus 1,"},{"Start":"03:25.020 ","End":"03:28.155","Text":"and here, square root of x plus 1."},{"Start":"03:28.155 ","End":"03:32.040","Text":"Now, I haven\u0027t changed anything because I\u0027ve multiplied by 1."},{"Start":"03:33.470 ","End":"03:43.900","Text":"In the denominator, this times this is just x minus 1,"},{"Start":"03:45.950 ","End":"03:48.915","Text":"because of this squared minus this squared."},{"Start":"03:48.915 ","End":"03:54.755","Text":"On the numerator, we have x minus 1, which is here,"},{"Start":"03:54.755 ","End":"04:04.770","Text":"times square root of x plus 1."},{"Start":"04:04.770 ","End":"04:08.620","Text":"Now, this cancels with this."},{"Start":"04:10.160 ","End":"04:14.220","Text":"I should have written the square root here."},{"Start":"04:14.350 ","End":"04:16.520","Text":"All we\u0027re left with is this."},{"Start":"04:16.520 ","End":"04:17.660","Text":"When x goes to 1,"},{"Start":"04:17.660 ","End":"04:19.940","Text":"this goes to square root of 1 plus 1,"},{"Start":"04:19.940 ","End":"04:23.920","Text":"which is 1 plus 1, which is equal to 2."},{"Start":"04:23.920 ","End":"04:29.045","Text":"Here we have 3 on the right,"},{"Start":"04:29.045 ","End":"04:31.685","Text":"but on the left we have 2."},{"Start":"04:31.685 ","End":"04:36.560","Text":"These things are not equal to each other."},{"Start":"04:36.560 ","End":"04:40.430","Text":"Because the left limit and the right limit are not the same,"},{"Start":"04:40.430 ","End":"04:43.290","Text":"the function has no limit."},{"Start":"04:43.820 ","End":"04:47.415","Text":"There is, I\u0027ll just write it in short as,"},{"Start":"04:47.415 ","End":"04:53.740","Text":"no limit, and leave that as the answer."}],"ID":1593},{"Watched":false,"Name":"Exercise 3","Duration":"3m 46s","ChapterTopicVideoID":1582,"CourseChapterTopicPlaylistID":170,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.160","Text":"In this exercise, we have to find the limit as x goes to 0 of absolute value of x over x."},{"Start":"00:08.160 ","End":"00:11.650","Text":"Let\u0027s call this function f of x."},{"Start":"00:12.830 ","End":"00:17.939","Text":"We have that f of x is equal"},{"Start":"00:17.939 ","End":"00:24.645","Text":"to absolute value of x over x,"},{"Start":"00:24.645 ","End":"00:27.405","Text":"and then we\u0027re asking,"},{"Start":"00:27.405 ","End":"00:40.685","Text":"what is the limit as x goes to 0 of f of x equals what?"},{"Start":"00:40.685 ","End":"00:45.320","Text":"Just wanted to use the notation f to give our function a name."},{"Start":"00:45.320 ","End":"00:52.145","Text":"Now, because absolute value of x is a function which is defined piecewise,"},{"Start":"00:52.145 ","End":"00:57.864","Text":"1 for when x is less than 0 and 1 for x greater or equal to 0."},{"Start":"00:57.864 ","End":"01:01.050","Text":"What I can say is that,"},{"Start":"01:01.050 ","End":"01:06.085","Text":"I\u0027ll first take the limit on the left and say that the limit,"},{"Start":"01:06.085 ","End":"01:10.970","Text":"or let\u0027s take the right first."},{"Start":"01:10.970 ","End":"01:17.480","Text":"x goes to 0 plus of f of x is equal"},{"Start":"01:17.480 ","End":"01:24.855","Text":"to the limit as x goes to 0 plus."},{"Start":"01:24.855 ","End":"01:29.779","Text":"Now, if x goes from the right, x is positive,"},{"Start":"01:29.779 ","End":"01:31.685","Text":"and when x is positive,"},{"Start":"01:31.685 ","End":"01:35.065","Text":"absolute value of x is just equal to x,"},{"Start":"01:35.065 ","End":"01:38.500","Text":"so it\u0027s x over x."},{"Start":"01:38.500 ","End":"01:42.930","Text":"Which is this over this is 1,"},{"Start":"01:42.930 ","End":"01:50.480","Text":"so the limit as x goes to 0 on the right or otherwise, of x over x is just 1."},{"Start":"01:50.480 ","End":"01:53.680","Text":"It\u0027s the limit of 1, so it\u0027s equal to 1."},{"Start":"01:53.680 ","End":"02:02.300","Text":"On the other hand, the limit as x goes to 0 from the left of"},{"Start":"02:02.300 ","End":"02:10.999","Text":"f of x is equal to the limit as x goes to 0 from the left."},{"Start":"02:10.999 ","End":"02:14.690","Text":"Now, here, when x is less than 0,"},{"Start":"02:14.690 ","End":"02:16.550","Text":"because it\u0027s going to 0 from the left,"},{"Start":"02:16.550 ","End":"02:17.960","Text":"x is less than 0,"},{"Start":"02:17.960 ","End":"02:21.455","Text":"and when x is less than 0,"},{"Start":"02:21.455 ","End":"02:27.140","Text":"then absolute value of x is equal to minus x"},{"Start":"02:27.140 ","End":"02:39.510","Text":"and this equals whatever x is."},{"Start":"02:39.510 ","End":"02:42.560","Text":"If it\u0027s negative, this over this is minus 1,"},{"Start":"02:42.560 ","End":"02:46.080","Text":"so the limit is also minus 1."},{"Start":"02:46.160 ","End":"02:52.505","Text":"What we have is that the limit as x goes to 0 from the right is 1."},{"Start":"02:52.505 ","End":"02:54.905","Text":"But as it goes to 0 from the left,"},{"Start":"02:54.905 ","End":"02:57.175","Text":"we have minus 1."},{"Start":"02:57.175 ","End":"02:59.330","Text":"Because these 2 are different,"},{"Start":"02:59.330 ","End":"03:04.100","Text":"then the function itself has no limit"},{"Start":"03:04.100 ","End":"03:08.375","Text":"when x goes to 0 because the left is different from the right."},{"Start":"03:08.375 ","End":"03:13.031","Text":"These two are not equal."},{"Start":"03:15.031 ","End":"03:24.570","Text":"Or if we like, the absolute value of x over x"},{"Start":"03:24.570 ","End":"03:39.635","Text":"has no limit as x goes to 0."},{"Start":"03:39.635 ","End":"03:44.210","Text":"It has a left limit and a right limit, but no limit."},{"Start":"03:44.210 ","End":"03:47.670","Text":"So no limit is the answer. We\u0027re done."}],"ID":1594},{"Watched":false,"Name":"Exercise 4","Duration":"44s","ChapterTopicVideoID":1583,"CourseChapterTopicPlaylistID":170,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:03.870","Text":"In this exercise, we have to find the limit as x goes"},{"Start":"00:03.870 ","End":"00:07.425","Text":"to infinity of absolute value of x over x."},{"Start":"00:07.425 ","End":"00:09.374","Text":"Now, when x turns to infinity,"},{"Start":"00:09.374 ","End":"00:11.325","Text":"x is certainly positive."},{"Start":"00:11.325 ","End":"00:13.155","Text":"If x is positive,"},{"Start":"00:13.155 ","End":"00:16.020","Text":"then the absolute value of x is just x."},{"Start":"00:16.020 ","End":"00:22.530","Text":"What we actually get is the limit as x goes to infinity,"},{"Start":"00:22.530 ","End":"00:23.895","Text":"this thing equals this."},{"Start":"00:23.895 ","End":"00:28.725","Text":"X goes to infinity of x over x,"},{"Start":"00:28.725 ","End":"00:30.225","Text":"because x is positive,"},{"Start":"00:30.225 ","End":"00:38.280","Text":"which is just the limit as x goes to infinity of 1,"},{"Start":"00:38.280 ","End":"00:40.740","Text":"and this is equal to 1."},{"Start":"00:40.740 ","End":"00:44.860","Text":"That\u0027s it. That\u0027s all there is to it."}],"ID":1595},{"Watched":false,"Name":"Exercise 5","Duration":"51s","ChapterTopicVideoID":1584,"CourseChapterTopicPlaylistID":170,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:04.020 ","End":"00:09.060","Text":"minus infinity of absolute value of x over x."},{"Start":"00:09.060 ","End":"00:13.650","Text":"Now, if x goes to minus infinity, it\u0027s certainly negative."},{"Start":"00:13.650 ","End":"00:15.660","Text":"When x is negative,"},{"Start":"00:15.660 ","End":"00:20.040","Text":"the absolute value of x we know is minus x."},{"Start":"00:20.040 ","End":"00:29.025","Text":"What this thing equals is just the limit of x goes to minus infinity,"},{"Start":"00:29.025 ","End":"00:33.915","Text":"absolute value of x is minus x over x,"},{"Start":"00:33.915 ","End":"00:36.400","Text":"which is the limit."},{"Start":"00:36.400 ","End":"00:39.890","Text":"As x goes to minus infinity,"},{"Start":"00:39.890 ","End":"00:42.025","Text":"this over this is minus 1."},{"Start":"00:42.025 ","End":"00:45.800","Text":"The limit of a constant is just the constant itself,"},{"Start":"00:45.800 ","End":"00:48.590","Text":"so the answer is minus 1."},{"Start":"00:48.590 ","End":"00:51.270","Text":"That\u0027s all, we\u0027re done."}],"ID":1596}],"Thumbnail":null,"ID":170}]

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