Function of Several Variables
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- Functions of two variables
- The Limit of a Function of Two Variables
- Polar Substitution
- Iterated Limits
- Limit along a Path
- Choosing the Right Technique
- Continuity of a Function of Two Variables
- Exercise 1 part a
- Exercise 1 part b
- Exercise 1 part c
- Exercise 1 part d
- Exercise 1 part e
- Exercise 1 part f
- Exercise 1 part g
- Exercise 1 part h
- Exercise 2 part a
- Exercise 2 part b
- Exercise 2 part c
- Exercise 2 part d
- Exercise 2 part e
- Exercise 2 part f
- Exercise 2 part g
- Exercise 2 part h
- Exercise 3 part a
- Exercise 3 part b
- Exercise 3 part c
- Exercise 3 part d
- Exercise 3 part e
- Exercise 3 part f
- Exercise 3 part g
- Exercise 3 part h
- Exercise 4 part a
- Exercise 4 part b
- Exercise 4 part c
- Exercise 4 part d
- Exercise 4 part e
- Exercise 4 part f
- Exercise 4 part g
- Exercise 4 part h
- Exercise 5 part a
- Exercise 5 part b

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[{"Name":"Function of Several Variables","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Functions of two variables","Duration":"17m 34s","ChapterTopicVideoID":8507,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8507.jpeg","UploadDate":"2020-02-22T22:10:43.3900000","DurationForVideoObject":"PT17M34S","Description":null,"MetaTitle":"Functions of two variables: Video + Workbook | Proprep","MetaDescription":"Functions of Several Variables - Function of Several Variables. 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/functions-of-several-variables/function-of-several-variables/vid8881","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"Up to now we\u0027ve been studying functions of a single variable,"},{"Start":"00:04.185 ","End":"00:07.800","Text":"and in this clip I\u0027ll introduce functions of 2 variables."},{"Start":"00:07.800 ","End":"00:10.005","Text":"Let\u0027s start with a story."},{"Start":"00:10.005 ","End":"00:18.000","Text":"Danny goes to the store to buy x pounds of tomatoes,"},{"Start":"00:18.000 ","End":"00:20.970","Text":"that\u0027s tomatoes to the Americans,"},{"Start":"00:20.970 ","End":"00:26.800","Text":"and y pounds of cucumbers."},{"Start":"00:26.800 ","End":"00:33.400","Text":"Now, tomatoes cost 50 cents a pound,"},{"Start":"00:34.310 ","End":"00:43.450","Text":"and cucumbers cost only 40 cents a pound."},{"Start":"00:44.270 ","End":"00:48.755","Text":"The question is, how much did Danny pay?"},{"Start":"00:48.755 ","End":"00:52.265","Text":"I mean, the total for the tomatoes and cucumbers."},{"Start":"00:52.265 ","End":"00:55.835","Text":"First of all, I should decide if I\u0027m going to work in dollars or cents."},{"Start":"00:55.835 ","End":"00:59.980","Text":"Let\u0027s say that I\u0027m going to work in dollars."},{"Start":"00:59.980 ","End":"01:08.430","Text":"I guess this is $0.5 and this is $0.4,"},{"Start":"01:09.080 ","End":"01:13.640","Text":"so we\u0027ll work in dollars and let\u0027s compute"},{"Start":"01:13.640 ","End":"01:18.140","Text":"separately the tomatoes then the cucumbers and we add them."},{"Start":"01:18.140 ","End":"01:24.020","Text":"So x pounds of tomatoes means that he paid"},{"Start":"01:24.020 ","End":"01:32.945","Text":"0.5 times x because it\u0027s the price per pound times the number of pounds."},{"Start":"01:32.945 ","End":"01:34.850","Text":"For the cucumber he paid,"},{"Start":"01:34.850 ","End":"01:41.715","Text":"the price per pound is 0.4 and the number of pounds is y,"},{"Start":"01:41.715 ","End":"01:43.820","Text":"and if we add these together,"},{"Start":"01:43.820 ","End":"01:47.525","Text":"that\u0027s the expression for what he paid."},{"Start":"01:47.525 ","End":"01:52.685","Text":"Notice this is an expression or a function of 2 variables, x and y."},{"Start":"01:52.685 ","End":"01:56.345","Text":"We use the notation f, and we put x,"},{"Start":"01:56.345 ","End":"02:00.140","Text":"y to indicate it\u0027s a function of 2 variables,"},{"Start":"02:00.140 ","End":"02:01.895","Text":"and this is what it equals."},{"Start":"02:01.895 ","End":"02:07.130","Text":"Let\u0027s substitute some values and get a better idea of this function of 2 variables."},{"Start":"02:07.130 ","End":"02:11.390","Text":"Suppose I want to know how much does it cost to"},{"Start":"02:11.390 ","End":"02:15.875","Text":"buy 1 pound of tomatoes and 2 pounds of cucumbers."},{"Start":"02:15.875 ","End":"02:19.620","Text":"So that means that x is 1 and y is 2."},{"Start":"02:19.620 ","End":"02:21.635","Text":"So this time when I substitute,"},{"Start":"02:21.635 ","End":"02:23.630","Text":"I mean, as opposed to a single variable,"},{"Start":"02:23.630 ","End":"02:25.809","Text":"I have to substitute 2 numbers,"},{"Start":"02:25.809 ","End":"02:28.555","Text":"I have to substitute x and y."},{"Start":"02:28.555 ","End":"02:30.850","Text":"This is the 1 pound of tomato,"},{"Start":"02:30.850 ","End":"02:33.115","Text":"this is 2 pounds of cucumbers."},{"Start":"02:33.115 ","End":"02:36.470","Text":"I don\u0027t have to go back to the story to figure it out,"},{"Start":"02:36.470 ","End":"02:41.300","Text":"I just plug it in this function here of 2 variables and I\u0027ll get the answer."},{"Start":"02:41.300 ","End":"02:49.160","Text":"I say that it\u0027s 0.5 times 1 plus 0.4 times"},{"Start":"02:49.160 ","End":"02:59.775","Text":"2 and altogether I get $1.3 or now we usually put the 0."},{"Start":"02:59.775 ","End":"03:01.565","Text":"Another example."},{"Start":"03:01.565 ","End":"03:09.260","Text":"How much does it cost to buy 0 pounds of tomatoes and 0 pounds of cucumbers?"},{"Start":"03:09.260 ","End":"03:12.935","Text":"Well, if I substitute x as 0 and y as 0,"},{"Start":"03:12.935 ","End":"03:15.230","Text":"clearly I get 0 and that makes sense."},{"Start":"03:15.230 ","End":"03:18.310","Text":"If I didn\u0027t buy any, it didn\u0027t cost anything."},{"Start":"03:18.310 ","End":"03:22.940","Text":"Yet another example, how much does it cost to buy"},{"Start":"03:22.940 ","End":"03:29.270","Text":"4 pounds of tomatoes and 7 pounds of cucumbers."},{"Start":"03:29.270 ","End":"03:31.595","Text":"What I get is again,"},{"Start":"03:31.595 ","End":"03:33.410","Text":"plugging in this function here,"},{"Start":"03:33.410 ","End":"03:35.105","Text":"the function of 2 variables,"},{"Start":"03:35.105 ","End":"03:43.885","Text":"we get 0.5 times 4 plus 0.4 times 7,"},{"Start":"03:43.885 ","End":"03:49.450","Text":"and this equals, I make it 4.8."},{"Start":"03:49.580 ","End":"03:58.055","Text":"Of course at the end I just write it in dollars and I add an extra 0, so $4.80."},{"Start":"03:58.055 ","End":"04:01.090","Text":"I think you get the idea."},{"Start":"04:01.090 ","End":"04:03.555","Text":"But let me ask this."},{"Start":"04:03.555 ","End":"04:09.285","Text":"What is f of minus 1 and 7,"},{"Start":"04:09.285 ","End":"04:13.140","Text":"then I would have to say undefined."},{"Start":"04:13.140 ","End":"04:15.425","Text":"Why is that?"},{"Start":"04:15.425 ","End":"04:20.165","Text":"Certainly I could substitute minus 1 and 7 here,"},{"Start":"04:20.165 ","End":"04:26.660","Text":"but it doesn\u0027t make sense to buy a negative quantity of tomatoes."},{"Start":"04:26.660 ","End":"04:30.015","Text":"I can\u0027t by negative amount,"},{"Start":"04:30.015 ","End":"04:33.165","Text":"and just like functions with 1 variable,"},{"Start":"04:33.165 ","End":"04:36.995","Text":"functions in 2 variables also have a domain of definition."},{"Start":"04:36.995 ","End":"04:42.725","Text":"In this case, clearly it\u0027s x bigger or equal to 0,"},{"Start":"04:42.725 ","End":"04:46.065","Text":"and y bigger or equal to 0."},{"Start":"04:46.065 ","End":"04:48.680","Text":"Quantities, they have to be positive."},{"Start":"04:48.680 ","End":"04:52.535","Text":"In general, when we have a function of 2 variables,"},{"Start":"04:52.535 ","End":"04:57.740","Text":"we can look at it as a function that takes in a point."},{"Start":"04:57.740 ","End":"05:00.720","Text":"After all, a pair of values is like a point in the plane."},{"Start":"05:00.720 ","End":"05:02.810","Text":"In this case, it takes the point 1,"},{"Start":"05:02.810 ","End":"05:08.570","Text":"2 and gives back a number 1.3."},{"Start":"05:08.570 ","End":"05:10.970","Text":"It takes in 0,"},{"Start":"05:10.970 ","End":"05:16.400","Text":"0 and gives out 0."},{"Start":"05:16.400 ","End":"05:18.830","Text":"It takes in 4,"},{"Start":"05:18.830 ","End":"05:24.950","Text":"7, and gives out 4.8 and so on."},{"Start":"05:24.950 ","End":"05:27.515","Text":"That\u0027s in general function in 2 variables,"},{"Start":"05:27.515 ","End":"05:30.565","Text":"it gets a point in the plane,"},{"Start":"05:30.565 ","End":"05:35.690","Text":"that\u0027s a pair of values and gives back a single value."},{"Start":"05:35.690 ","End":"05:39.710","Text":"Just like in the case of a function of 1 variable,"},{"Start":"05:39.710 ","End":"05:41.940","Text":"we gave a letter for the output also."},{"Start":"05:41.940 ","End":"05:46.400","Text":"We would say y equals f of x, in 2 variables,"},{"Start":"05:46.400 ","End":"05:49.585","Text":"the output is usually the letter,"},{"Start":"05:49.585 ","End":"05:53.805","Text":"Z or Z if you\u0027re in America."},{"Start":"05:53.805 ","End":"05:59.600","Text":"We say that Z equals f of x, y,"},{"Start":"05:59.600 ","End":"06:06.235","Text":"sometimes I cross the Z to avoid confusing with the digit 2."},{"Start":"06:06.235 ","End":"06:09.845","Text":"I also can write Z sometimes like this."},{"Start":"06:09.845 ","End":"06:13.250","Text":"Anyway, this will be Z or Z if you\u0027re in England."},{"Start":"06:13.250 ","End":"06:19.470","Text":"Let me give a definition of a function of 2 variables."},{"Start":"06:19.700 ","End":"06:25.475","Text":"A function of 2 variables is a rule or formula"},{"Start":"06:25.475 ","End":"06:31.740","Text":"which assigns to each point x,"},{"Start":"06:31.740 ","End":"06:34.890","Text":"y, a point is a pair of numbers."},{"Start":"06:34.890 ","End":"06:41.120","Text":"In a domain, I write this because it doesn\u0027t mean every point x, y will do."},{"Start":"06:41.120 ","End":"06:43.010","Text":"It might be restricted."},{"Start":"06:43.010 ","End":"06:46.175","Text":"So it assigns to each point a value."},{"Start":"06:46.175 ","End":"06:49.145","Text":"In fact, we\u0027ll call that value Z."},{"Start":"06:49.145 ","End":"06:52.940","Text":"There we are. Here we have an example."},{"Start":"06:52.940 ","End":"07:00.920","Text":"We also say that we substitute x and y into the function and get Z."},{"Start":"07:00.920 ","End":"07:05.780","Text":"I want to add that the set of values that were allowed to"},{"Start":"07:05.780 ","End":"07:10.980","Text":"substitute is what we call the domain of definition."},{"Start":"07:12.210 ","End":"07:15.745","Text":"We\u0027ve defined the domain of definition."},{"Start":"07:15.745 ","End":"07:18.340","Text":"It\u0027s the values we\u0027re allowed to substitute"},{"Start":"07:18.340 ","End":"07:21.715","Text":"in the function and in our particular example,"},{"Start":"07:21.715 ","End":"07:24.520","Text":"it was the points x,"},{"Start":"07:24.520 ","End":"07:29.845","Text":"y where x is bigger or equal to 0 and y bigger or equal to 0."},{"Start":"07:29.845 ","End":"07:35.995","Text":"That is what I meant by inner domain here in the definition."},{"Start":"07:35.995 ","End":"07:40.330","Text":"To each point but the points have to be or may be restricted,"},{"Start":"07:40.330 ","End":"07:41.980","Text":"of course it could be everything."},{"Start":"07:41.980 ","End":"07:45.310","Text":"I underline this also because we\u0027ve made 2 definitions here,"},{"Start":"07:45.310 ","End":"07:49.015","Text":"a function of 2 variables and the domain of definition."},{"Start":"07:49.015 ","End":"07:53.110","Text":"Sometimes we can graphically illustrate the domain of definition,"},{"Start":"07:53.110 ","End":"07:54.340","Text":"it might be useful."},{"Start":"07:54.340 ","End":"07:56.080","Text":"For example, in our case,"},{"Start":"07:56.080 ","End":"07:58.630","Text":"if I want to sketch the domain,"},{"Start":"07:58.630 ","End":"08:00.850","Text":"let me just do this just very roughly."},{"Start":"08:00.850 ","End":"08:03.400","Text":"Here\u0027s y, here\u0027s x."},{"Start":"08:03.400 ","End":"08:07.960","Text":"This side of the x-axis,"},{"Start":"08:07.960 ","End":"08:11.350","Text":"this side is where x is bigger than"},{"Start":"08:11.350 ","End":"08:15.550","Text":"0 and above this axis is where y is bigger or equal to 0."},{"Start":"08:15.550 ","End":"08:18.834","Text":"Altogether, where both are bigger than 0,"},{"Start":"08:18.834 ","End":"08:21.385","Text":"then this is the,"},{"Start":"08:21.385 ","End":"08:26.290","Text":"it\u0027s oldest, what we call the first quadrant."},{"Start":"08:26.290 ","End":"08:34.120","Text":"This is the domain where x is bigger or equal to 0 and y bigger or equal to 0."},{"Start":"08:34.120 ","End":"08:37.750","Text":"But we don\u0027t always want to or need to illustrate,"},{"Start":"08:37.750 ","End":"08:39.130","Text":"I\u0027m just saying we can."},{"Start":"08:39.130 ","End":"08:42.565","Text":"Now, let\u0027s take another example of a function."},{"Start":"08:42.565 ","End":"08:46.105","Text":"This time I\u0027ll use a different letter,"},{"Start":"08:46.105 ","End":"08:50.500","Text":"not f, I\u0027ll call it g. Oops, wrong color."},{"Start":"08:50.500 ","End":"08:54.714","Text":"G of x, y,"},{"Start":"08:54.714 ","End":"08:57.160","Text":"also a function in 2 variables,"},{"Start":"08:57.160 ","End":"09:03.980","Text":"is equal to the square root of x plus y squared."},{"Start":"09:04.890 ","End":"09:15.070","Text":"For example, g of the point 1,2 is equal to the square root of 1 plus 2^2,"},{"Start":"09:15.070 ","End":"09:18.220","Text":"which is equal to 5."},{"Start":"09:18.220 ","End":"09:20.500","Text":"In this example also,"},{"Start":"09:20.500 ","End":"09:22.630","Text":"we have a domain of definition."},{"Start":"09:22.630 ","End":"09:30.805","Text":"If you look at it, we see that the domain is just x bigger or equal to 0."},{"Start":"09:30.805 ","End":"09:33.460","Text":"Because only the square root might be a problem,"},{"Start":"09:33.460 ","End":"09:34.840","Text":"y could be anything."},{"Start":"09:34.840 ","End":"09:37.870","Text":"We could leave it like this or you could write"},{"Start":"09:37.870 ","End":"09:43.780","Text":"all y if you want to be explicit that y is not restricted."},{"Start":"09:43.780 ","End":"09:47.379","Text":"Let\u0027s just take another example of values."},{"Start":"09:47.379 ","End":"09:50.140","Text":"X has to be non-negative,"},{"Start":"09:50.140 ","End":"09:51.580","Text":"but y could be negative."},{"Start":"09:51.580 ","End":"10:00.025","Text":"We could put minus 4 for y and then we would get square root of 4 plus minus 4^2,"},{"Start":"10:00.025 ","End":"10:02.770","Text":"and that would be 18."},{"Start":"10:02.770 ","End":"10:07.750","Text":"Let\u0027s move on, take another example of a function in 2 variables."},{"Start":"10:07.750 ","End":"10:14.845","Text":"Let h of x, y equal x^2."},{"Start":"10:14.845 ","End":"10:19.495","Text":"Notice that y doesn\u0027t have to explicitly appear here."},{"Start":"10:19.495 ","End":"10:22.524","Text":"Just like in functions of 1 variable,"},{"Start":"10:22.524 ","End":"10:26.680","Text":"we sometimes had f of x equals 3, a constant function."},{"Start":"10:26.680 ","End":"10:28.210","Text":"Well, this is not a constant function,"},{"Start":"10:28.210 ","End":"10:30.100","Text":"but it doesn\u0027t depend on y."},{"Start":"10:30.100 ","End":"10:35.410","Text":"For example, h of 1,2 is just x squared,"},{"Start":"10:35.410 ","End":"10:36.999","Text":"so it\u0027s 1 squared."},{"Start":"10:36.999 ","End":"10:39.130","Text":"There is nowhere to substitute this 2,"},{"Start":"10:39.130 ","End":"10:43.610","Text":"it\u0027s irrelevant for this function which is equal to 1."},{"Start":"10:43.860 ","End":"10:49.450","Text":"Another example, this time a function that depends only on y."},{"Start":"10:49.450 ","End":"10:57.760","Text":"Let\u0027s say z of x,"},{"Start":"10:57.760 ","End":"11:03.415","Text":"y is equal to y plus 5."},{"Start":"11:03.415 ","End":"11:09.370","Text":"For instance, z of 0,7."},{"Start":"11:09.370 ","End":"11:12.670","Text":"Ignore the zero, know where to put."},{"Start":"11:12.670 ","End":"11:17.769","Text":"It\u0027s just 7 plus 5, which equals 12."},{"Start":"11:17.769 ","End":"11:20.500","Text":"We take the point 0,"},{"Start":"11:20.500 ","End":"11:23.215","Text":"7, and we assign it to 12."},{"Start":"11:23.215 ","End":"11:25.060","Text":"I should have emphasized this before,"},{"Start":"11:25.060 ","End":"11:29.275","Text":"g takes the point 4 minus 4, assigns it to 18."},{"Start":"11:29.275 ","End":"11:35.045","Text":"It assigns 1, 2 to the number 5."},{"Start":"11:35.045 ","End":"11:40.250","Text":"H takes 1,2 and assigns it to 1."},{"Start":"11:40.250 ","End":"11:44.505","Text":"I\u0027m trying to just emphasize the idea that we take"},{"Start":"11:44.505 ","End":"11:49.200","Text":"a point which is a pair of numbers and assign to them a single number,"},{"Start":"11:49.200 ","End":"11:52.755","Text":"and that\u0027s how a function of 2 variables works."},{"Start":"11:52.755 ","End":"11:55.935","Text":"I should have mentioned the domain of these."},{"Start":"11:55.935 ","End":"12:00.490","Text":"The domain of h is just all x and y."},{"Start":"12:00.490 ","End":"12:01.930","Text":"Sometimes I don\u0027t write anything,"},{"Start":"12:01.930 ","End":"12:09.500","Text":"it means there is no restriction and the domain of z is also all of x and y."},{"Start":"12:09.840 ","End":"12:19.750","Text":"Next, I want to talk about the graph of a function in 2 variables."},{"Start":"12:19.750 ","End":"12:22.900","Text":"Let me emphasize that you will not be required"},{"Start":"12:22.900 ","End":"12:27.820","Text":"to draw graphs of a function of 2 variables on the exam."},{"Start":"12:27.820 ","End":"12:32.665","Text":"I\u0027m just using it for illustration purposes."},{"Start":"12:32.665 ","End":"12:36.235","Text":"You will have to possibly sketch something else."},{"Start":"12:36.235 ","End":"12:38.395","Text":"But let\u0027s leave that for now."},{"Start":"12:38.395 ","End":"12:39.775","Text":"But in any event, the graphs,"},{"Start":"12:39.775 ","End":"12:42.310","Text":"you will not need to be able to do,"},{"Start":"12:42.310 ","End":"12:44.215","Text":"I\u0027m just showing you."},{"Start":"12:44.215 ","End":"12:49.930","Text":"What we\u0027ll do is, we\u0027ll take the first example with Danny and the vegetables."},{"Start":"12:49.930 ","End":"12:56.800","Text":"Here we are in a new page and I copied the function from before with the story where"},{"Start":"12:56.800 ","End":"13:04.820","Text":"Danny buys x pounds of tomatoes and y pounds of cucumbers and this is the cost function."},{"Start":"13:05.280 ","End":"13:09.580","Text":"1 of the main problems of drawing a graph of a function of"},{"Start":"13:09.580 ","End":"13:13.705","Text":"2 variables is it has to be in 3-dimensions."},{"Start":"13:13.705 ","End":"13:17.590","Text":"Of course, we could do a 2-dimensional picture of"},{"Start":"13:17.590 ","End":"13:23.140","Text":"a 3-dimensional graph, but it\u0027s problematic."},{"Start":"13:23.140 ","End":"13:26.725","Text":"I\u0027ll show you why we need 3-dimensions."},{"Start":"13:26.725 ","End":"13:30.760","Text":"Consider first of all, a function of 1 variable."},{"Start":"13:30.760 ","End":"13:38.065","Text":"For example, let\u0027s just say we took f of x is equal to x^2,"},{"Start":"13:38.065 ","End":"13:42.100","Text":"or even y equals f of x equals x^2."},{"Start":"13:42.100 ","End":"13:45.265","Text":"We need a variable x,"},{"Start":"13:45.265 ","End":"13:48.730","Text":"and we need a variable for the function that we get."},{"Start":"13:48.730 ","End":"13:51.610","Text":"For example, when x is 0,"},{"Start":"13:51.610 ","End":"13:54.940","Text":"y is 0, when x is 1, y is 1,"},{"Start":"13:54.940 ","End":"13:58.840","Text":"or f of x is 1, minus 1,1,"},{"Start":"13:58.840 ","End":"14:02.665","Text":"minus 2,4, and so on."},{"Start":"14:02.665 ","End":"14:06.220","Text":"For each x we get a y and we get a point x,"},{"Start":"14:06.220 ","End":"14:09.550","Text":"y and all this takes place in the plane."},{"Start":"14:09.550 ","End":"14:12.590","Text":"I\u0027m going to erase that."},{"Start":"14:13.860 ","End":"14:20.665","Text":"What happens in 2 variables is that for each x and for each y,"},{"Start":"14:20.665 ","End":"14:22.480","Text":"we get f of x, y,"},{"Start":"14:22.480 ","End":"14:25.900","Text":"or we could call it z or z."},{"Start":"14:25.900 ","End":"14:29.000","Text":"What would happen is we need"},{"Start":"14:30.360 ","End":"14:37.630","Text":"2 variables for y and for x."},{"Start":"14:37.630 ","End":"14:39.589","Text":"For each x and y,"},{"Start":"14:39.589 ","End":"14:42.814","Text":"we get a value of z."},{"Start":"14:42.814 ","End":"14:50.015","Text":"For example, earlier, we actually found some points on the graph."},{"Start":"14:50.015 ","End":"14:57.635","Text":"We had that f of 1,2 was equal to 1.30."},{"Start":"14:57.635 ","End":"15:04.120","Text":"We also had f of 0,0 is equal to 0,"},{"Start":"15:04.120 ","End":"15:08.365","Text":"and we had f of 4,7,"},{"Start":"15:08.365 ","End":"15:13.840","Text":"which was equal to $4.80."},{"Start":"15:13.840 ","End":"15:17.375","Text":"To take a point like this and to plot it means,"},{"Start":"15:17.375 ","End":"15:19.700","Text":"for example here, x is 1,"},{"Start":"15:19.700 ","End":"15:22.490","Text":"so we go 1 unit along the x-axis,"},{"Start":"15:22.490 ","End":"15:29.075","Text":"y is 2 so we go 2 units along the y-axis."},{"Start":"15:29.075 ","End":"15:34.635","Text":"This brings us to the point 1,2,"},{"Start":"15:34.635 ","End":"15:39.640","Text":"and now this is the value of z."},{"Start":"15:39.640 ","End":"15:43.190","Text":"Z here is 1.3."},{"Start":"15:43.860 ","End":"15:46.390","Text":"Somewhere, let\u0027s say here,"},{"Start":"15:46.390 ","End":"15:56.355","Text":"is 1.3 and then we have to go up 1.3 somewhere like here."},{"Start":"15:56.355 ","End":"15:58.540","Text":"It\u0027s hard to say."},{"Start":"15:58.650 ","End":"16:03.620","Text":"In general, when I\u0027ve started plotting more of these points,"},{"Start":"16:03.620 ","End":"16:07.830","Text":"these points will all form some surface."},{"Start":"16:11.730 ","End":"16:16.735","Text":"In 3-dimensions we\u0027ll get a curve."},{"Start":"16:16.735 ","End":"16:19.615","Text":"Let me just show some more points."},{"Start":"16:19.615 ","End":"16:22.990","Text":"I have the 0,0, I have 4,7."},{"Start":"16:22.990 ","End":"16:26.230","Text":"I\u0027m not trying to be accurate or to scale or anything."},{"Start":"16:26.230 ","End":"16:27.925","Text":"I get the 4,7."},{"Start":"16:27.925 ","End":"16:31.410","Text":"From here, I go up to 4.8."},{"Start":"16:31.410 ","End":"16:33.940","Text":"Well, we can see the whole thing\u0027s not accurate but in general,"},{"Start":"16:33.940 ","End":"16:36.400","Text":"the idea is to get several points."},{"Start":"16:36.400 ","End":"16:39.670","Text":"Each time you can substitute a point and get a value,"},{"Start":"16:39.670 ","End":"16:42.040","Text":"and then you\u0027ll get a lot of these points and then you draw"},{"Start":"16:42.040 ","End":"16:45.550","Text":"a surface through and it\u0027s really a nightmare."},{"Start":"16:45.550 ","End":"16:51.450","Text":"As I said, you are not going to be required to sketch graphs in 3-dimensions,"},{"Start":"16:51.450 ","End":"16:54.345","Text":"graphs of functions of 2 variables."},{"Start":"16:54.345 ","End":"16:57.410","Text":"But this is just the idea of how it works."},{"Start":"16:57.410 ","End":"16:59.180","Text":"I\u0027m basically done here."},{"Start":"16:59.180 ","End":"17:02.900","Text":"But what I want to say is that even though it\u0027s impractical to"},{"Start":"17:02.900 ","End":"17:07.800","Text":"draw graphs except computer plotted images,"},{"Start":"17:07.930 ","End":"17:10.340","Text":"I\u0027ll just give you the names now,"},{"Start":"17:10.340 ","End":"17:12.530","Text":"but that\u0027s for the future."},{"Start":"17:12.530 ","End":"17:15.785","Text":"There are things called contour lines,"},{"Start":"17:15.785 ","End":"17:21.410","Text":"and these you will learn how to plot in a future clip."},{"Start":"17:21.410 ","End":"17:26.300","Text":"These are called in economics, level curves."},{"Start":"17:26.300 ","End":"17:30.980","Text":"It\u0027s the next best thing that you can do to drawing a graph."},{"Start":"17:30.980 ","End":"17:32.930","Text":"But that\u0027s not for now."},{"Start":"17:32.930 ","End":"17:35.340","Text":"Here, I\u0027m done."}],"ID":8881},{"Watched":false,"Name":"The Limit of a Function of Two Variables","Duration":"13m 32s","ChapterTopicVideoID":8508,"CourseChapterTopicPlaylistID":4970,"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 clip, I\u0027ll be talking about the limit of a function of 2 variables."},{"Start":"00:05.565 ","End":"00:09.690","Text":"We already know all about limits of a function of 1 variable."},{"Start":"00:09.690 ","End":"00:11.460","Text":"You learned that in Calculus I,"},{"Start":"00:11.460 ","End":"00:13.530","Text":"but now we\u0027re in Calculus II."},{"Start":"00:13.530 ","End":"00:16.245","Text":"We\u0027ll move on to 2 variables."},{"Start":"00:16.245 ","End":"00:19.170","Text":"Let\u0027s start with an example."},{"Start":"00:19.170 ","End":"00:22.995","Text":"Let\u0027s take a function of 2 variables,"},{"Start":"00:22.995 ","End":"00:25.260","Text":"which will be f of x,y,"},{"Start":"00:25.260 ","End":"00:30.585","Text":"which is equal to the square root of 2x plus"},{"Start":"00:30.585 ","End":"00:38.475","Text":"y minus 2 over 4x plus 2y minus 8."},{"Start":"00:38.475 ","End":"00:46.825","Text":"Notice that if I substitute x equals 1 and y equals 2,"},{"Start":"00:46.825 ","End":"00:52.440","Text":"then I get the square root of 4 minus 2 is 0."},{"Start":"00:52.440 ","End":"00:56.820","Text":"Here I get 8 minus 8 is also 0."},{"Start":"00:56.820 ","End":"00:59.539","Text":"We get 0 over 0, which is undefined."},{"Start":"00:59.539 ","End":"01:05.205","Text":"In other words, f of 1,2 is undefined."},{"Start":"01:05.205 ","End":"01:08.344","Text":"Clearly, the point 1,2 is not in the domain."},{"Start":"01:08.344 ","End":"01:11.840","Text":"Even though f of 1,2 is undefined,"},{"Start":"01:11.840 ","End":"01:15.220","Text":"it\u0027s legitimate to ask the following question."},{"Start":"01:15.220 ","End":"01:18.605","Text":"To what value, if any,"},{"Start":"01:18.605 ","End":"01:25.415","Text":"does f of x and y approach when x approaches 1 and y approaches 2?"},{"Start":"01:25.415 ","End":"01:29.690","Text":"This is the kind of question we asked when we studied limits of 1 variable."},{"Start":"01:29.690 ","End":"01:32.420","Text":"The value at the point maybe was undefined,"},{"Start":"01:32.420 ","End":"01:35.150","Text":"but then we did the next best thing of asking,"},{"Start":"01:35.150 ","End":"01:38.000","Text":"what does the value of the function get near when"},{"Start":"01:38.000 ","End":"01:40.970","Text":"the point gets near the place where it\u0027s undefined?"},{"Start":"01:40.970 ","End":"01:43.055","Text":"This is exactly what we\u0027re doing here."},{"Start":"01:43.055 ","End":"01:46.985","Text":"At exactly 1,2, we don\u0027t have a value of the function."},{"Start":"01:46.985 ","End":"01:50.540","Text":"We can ask if x,y gets very near 1,2,"},{"Start":"01:50.540 ","End":"01:54.365","Text":"perhaps f of x,y gets very near some value."},{"Start":"01:54.365 ","End":"01:56.390","Text":"I\u0027m going to rephrase this question in"},{"Start":"01:56.390 ","End":"01:59.585","Text":"terms of limits and then you\u0027ll see what I mean by limit."},{"Start":"01:59.585 ","End":"02:02.060","Text":"This is equivalent to saying,"},{"Start":"02:02.060 ","End":"02:07.020","Text":"what is the limit when x goes to"},{"Start":"02:07.020 ","End":"02:12.750","Text":"1 and y goes to 2 of f of x,y?"},{"Start":"02:12.750 ","End":"02:15.150","Text":"But I\u0027ll copy this down here."},{"Start":"02:15.150 ","End":"02:19.200","Text":"What is the limit when x goes to 1 and y goes to 2 of square root of"},{"Start":"02:19.200 ","End":"02:24.345","Text":"2x plus y minus 2 over 4x plus 2y minus 8?"},{"Start":"02:24.345 ","End":"02:27.955","Text":"This is how we use the concept of limit,"},{"Start":"02:27.955 ","End":"02:35.695","Text":"is to just write precisely what we loosely talk about when we say approach and approach."},{"Start":"02:35.695 ","End":"02:37.915","Text":"There\u0027s another way of writing this."},{"Start":"02:37.915 ","End":"02:40.809","Text":"Actually, the second way I even prefer,"},{"Start":"02:40.809 ","End":"02:44.940","Text":"because this notation implies that somehow x is getting"},{"Start":"02:44.940 ","End":"02:49.140","Text":"close to 1 and y getting close to 2, but perhaps separately."},{"Start":"02:49.140 ","End":"02:54.620","Text":"What I want is for the point to approach the point 1,2 as a point."},{"Start":"02:54.620 ","End":"02:57.305","Text":"There is another way of writing this."},{"Start":"02:57.305 ","End":"02:59.150","Text":"Let me erase this."},{"Start":"02:59.150 ","End":"03:08.815","Text":"We write that x,y approaches 1,2."},{"Start":"03:08.815 ","End":"03:11.190","Text":"The general point in the plane,"},{"Start":"03:11.190 ","End":"03:13.725","Text":"as it gets nearer to 1,2."},{"Start":"03:13.725 ","End":"03:20.010","Text":"We\u0027re going to learn how to compute such limits."},{"Start":"03:20.010 ","End":"03:22.635","Text":"There are a few techniques,"},{"Start":"03:22.635 ","End":"03:27.395","Text":"and the first 1 we\u0027re going to learn is the technique of substitution."},{"Start":"03:27.395 ","End":"03:32.065","Text":"Technique number 1, substitution."},{"Start":"03:32.065 ","End":"03:36.370","Text":"I\u0027ll just use this example,"},{"Start":"03:36.370 ","End":"03:38.360","Text":"and I\u0027ll show you what I mean by substitution."},{"Start":"03:38.360 ","End":"03:43.080","Text":"Now, those of you who are sharp eyed will notice that"},{"Start":"03:43.080 ","End":"03:48.030","Text":"4x plus 2y is exactly double 2x plus"},{"Start":"03:48.030 ","End":"03:56.015","Text":"y. I can write this as the limit of the square root of"},{"Start":"03:56.015 ","End":"04:00.049","Text":"2x plus y minus 2"},{"Start":"04:00.049 ","End":"04:06.780","Text":"over twice 2x plus y minus 8."},{"Start":"04:06.780 ","End":"04:09.705","Text":"I\u0027ve highlighted this 2x plus y,"},{"Start":"04:09.705 ","End":"04:13.670","Text":"and this is what I\u0027m going to substitute. I\u0027ll take another letter."},{"Start":"04:13.670 ","End":"04:20.585","Text":"Usually, we use the letter t. Then I\u0027ll let t equals 2x plus y."},{"Start":"04:20.585 ","End":"04:24.725","Text":"The general idea is to get this to be a limit of 1 variable."},{"Start":"04:24.725 ","End":"04:26.915","Text":"Now, if I do this substitution,"},{"Start":"04:26.915 ","End":"04:33.050","Text":"I\u0027m going to get the limit of the square root of"},{"Start":"04:33.050 ","End":"04:42.850","Text":"t minus 2 over 2t minus 8."},{"Start":"04:42.850 ","End":"04:47.070","Text":"I think I will write the limit in."},{"Start":"04:48.260 ","End":"04:51.870","Text":"The question is, what about t?"},{"Start":"04:51.870 ","End":"04:53.495","Text":"Where does t go to?"},{"Start":"04:53.495 ","End":"04:57.550","Text":"Well, if x and y go to 1 and 2 respectively,"},{"Start":"04:57.550 ","End":"05:02.900","Text":"then 2x plus y is twice 1 plus 2 is 4."},{"Start":"05:02.900 ","End":"05:04.925","Text":"So t goes to 4."},{"Start":"05:04.925 ","End":"05:08.210","Text":"Notice again that if actually put t equals 4,"},{"Start":"05:08.210 ","End":"05:09.750","Text":"then square root of 4 is 2,"},{"Start":"05:09.750 ","End":"05:13.170","Text":"minus 2 is 0, and twice t is 8, minus 8 is 0."},{"Start":"05:13.170 ","End":"05:14.475","Text":"I\u0027ve got 0 over 0."},{"Start":"05:14.475 ","End":"05:18.530","Text":"In that respect, that problem of 0 over 0 is still there."},{"Start":"05:18.530 ","End":"05:24.110","Text":"But I do have the benefit of having just 1 variable and I can handle this limit."},{"Start":"05:24.110 ","End":"05:25.865","Text":"There\u0027s many ways of doing this,"},{"Start":"05:25.865 ","End":"05:29.930","Text":"but the quickest is just because it\u0027s 0 over 0,"},{"Start":"05:29.930 ","End":"05:32.075","Text":"is to use L\u0027Hopital\u0027s rule."},{"Start":"05:32.075 ","End":"05:37.580","Text":"We replace this by the limit also as t goes to 4,"},{"Start":"05:37.580 ","End":"05:41.430","Text":"I\u0027ll just say here I\u0027m using L\u0027Hopital\u0027s rule."},{"Start":"05:42.530 ","End":"05:45.875","Text":"If I differentiate the numerator,"},{"Start":"05:45.875 ","End":"05:53.135","Text":"I get 1 over twice square root of t. On the denominator,"},{"Start":"05:53.135 ","End":"05:55.310","Text":"I just get 2."},{"Start":"05:55.310 ","End":"05:57.800","Text":"At this point there\u0027s no problem in just substituting."},{"Start":"05:57.800 ","End":"05:59.180","Text":"If I put t equals 4,"},{"Start":"05:59.180 ","End":"06:01.910","Text":"I get 1 over 2 times 2 over 2,"},{"Start":"06:01.910 ","End":"06:05.470","Text":"so it comes out to be 1/8."},{"Start":"06:05.470 ","End":"06:10.170","Text":"The answer to this limit is also 1/8."},{"Start":"06:10.210 ","End":"06:15.440","Text":"Let\u0027s go for another example of substitution technique."},{"Start":"06:15.440 ","End":"06:19.085","Text":"This time I\u0027ll take the limit."},{"Start":"06:19.085 ","End":"06:21.890","Text":"I\u0027ll use a different color so we don\u0027t confuse"},{"Start":"06:21.890 ","End":"06:25.355","Text":"the 2 examples or I\u0027ll put a little separating line here."},{"Start":"06:25.355 ","End":"06:32.000","Text":"This time it\u0027s the limit of the sine of 2x plus"},{"Start":"06:32.000 ","End":"06:41.475","Text":"2y over the sine of 10x plus 10y."},{"Start":"06:41.475 ","End":"06:43.680","Text":"Let\u0027s see. We\u0027ll take,"},{"Start":"06:43.680 ","End":"06:49.820","Text":"x,y goes to 0,0."},{"Start":"06:49.820 ","End":"06:54.995","Text":"Well, let\u0027s see what happens if we just substitute x,y as 0,0."},{"Start":"06:54.995 ","End":"06:57.980","Text":"If x and y are both 0, this comes out 0,"},{"Start":"06:57.980 ","End":"07:01.655","Text":"this comes out 0, we get sine 0 over sine 0."},{"Start":"07:01.655 ","End":"07:05.945","Text":"We have 1 of those 0 over 0 cases."},{"Start":"07:05.945 ","End":"07:09.605","Text":"But we don\u0027t have a L\u0027Hopital rule in 2 variables,"},{"Start":"07:09.605 ","End":"07:12.845","Text":"so let\u0027s see if we can use substitution."},{"Start":"07:12.845 ","End":"07:14.660","Text":"If you just look a bit at the function,"},{"Start":"07:14.660 ","End":"07:20.110","Text":"I think you\u0027ll immediately see that the thing to substitute would be,"},{"Start":"07:20.110 ","End":"07:24.890","Text":"well, take your pick, either x plus y or 2x plus 2y."},{"Start":"07:24.890 ","End":"07:27.695","Text":"I think we\u0027ll just go for x plus y."},{"Start":"07:27.695 ","End":"07:30.410","Text":"This is twice that, this is 10 times that."},{"Start":"07:30.410 ","End":"07:36.065","Text":"What we get is the limit of sine of"},{"Start":"07:36.065 ","End":"07:43.080","Text":"2t over sine of 10t."},{"Start":"07:43.080 ","End":"07:45.130","Text":"What does t go to?"},{"Start":"07:45.130 ","End":"07:47.840","Text":"Well, if t is x plus y and x goes to 0,"},{"Start":"07:47.840 ","End":"07:51.050","Text":"y goes to 0, then t also goes to 0."},{"Start":"07:51.050 ","End":"07:55.370","Text":"We still have the problem of sine 0 over sine 0."},{"Start":"07:55.370 ","End":"08:00.485","Text":"It\u0027s still 0 over 0, but this time it\u0027s in 1 variable."},{"Start":"08:00.485 ","End":"08:03.725","Text":"Again, I think I\u0027ll use the L\u0027Hopital rule."},{"Start":"08:03.725 ","End":"08:07.790","Text":"This gives us, if we differentiate the top and the bottom,"},{"Start":"08:07.790 ","End":"08:12.840","Text":"we get the new limit from L\u0027Hopital of twice"},{"Start":"08:12.840 ","End":"08:20.470","Text":"cosine 2t over 10 times cosine of 10t."},{"Start":"08:20.690 ","End":"08:25.020","Text":"Now, we can cancel the cosine."},{"Start":"08:25.020 ","End":"08:31.935","Text":"This is just equal to 2 over 10, which is 1/5."},{"Start":"08:31.935 ","End":"08:34.795","Text":"That\u0027s the answer. The limit of this is 1/5."},{"Start":"08:34.795 ","End":"08:36.320","Text":"How about another example?"},{"Start":"08:36.320 ","End":"08:38.135","Text":"We\u0027re doing so well."},{"Start":"08:38.135 ","End":"08:42.565","Text":"Let\u0027s take a third example or is it a fourth."},{"Start":"08:42.565 ","End":"08:44.034","Text":"We\u0027ll take the limit,"},{"Start":"08:44.034 ","End":"08:47.400","Text":"this time we\u0027ll go to 1,1."},{"Start":"08:47.400 ","End":"08:49.650","Text":"You know what, just for a change,"},{"Start":"08:49.650 ","End":"08:52.350","Text":"I\u0027ll write the limit the other way."},{"Start":"08:53.640 ","End":"08:57.070","Text":"I don\u0027t want you to get used to just writing it 1 way."},{"Start":"08:57.070 ","End":"09:00.700","Text":"Although I definitely prefer this notation,"},{"Start":"09:00.700 ","End":"09:02.050","Text":"but we\u0027ll write it this way,"},{"Start":"09:02.050 ","End":"09:04.060","Text":"x separately and y separately."},{"Start":"09:04.060 ","End":"09:12.680","Text":"I\u0027ve written the function for you already twice log x plus 4 log y over x^4 y^8 minus 1."},{"Start":"09:13.400 ","End":"09:17.980","Text":"At first sight, it doesn\u0027t seem to be anything to substitute."},{"Start":"09:17.980 ","End":"09:20.620","Text":"Let me go back."},{"Start":"09:20.620 ","End":"09:23.950","Text":"First of all, we should check there may not be any problem here."},{"Start":"09:23.950 ","End":"09:26.915","Text":"If I put x equals 1 and y equals 1,"},{"Start":"09:26.915 ","End":"09:30.585","Text":"natural log of 1 is 0."},{"Start":"09:30.585 ","End":"09:32.565","Text":"This is 0 and this is 0."},{"Start":"09:32.565 ","End":"09:34.860","Text":"What\u0027s more if these 2 are both 1,"},{"Start":"09:34.860 ","End":"09:38.070","Text":"then x^4 y^8 is also 1."},{"Start":"09:38.070 ","End":"09:42.135","Text":"Again, we have a 0 over 0 situation."},{"Start":"09:42.135 ","End":"09:44.765","Text":"I guess the idea is to substitute something,"},{"Start":"09:44.765 ","End":"09:47.670","Text":"make it a function in 1 variable."},{"Start":"09:47.670 ","End":"09:50.030","Text":"We\u0027ll still get 0 over 0,"},{"Start":"09:50.030 ","End":"09:55.844","Text":"but we\u0027ll only have 1 variable t. It doesn\u0027t look immediately obvious,"},{"Start":"09:55.844 ","End":"10:00.990","Text":"but I\u0027d like to remind you of some rules of logarithms."},{"Start":"10:00.990 ","End":"10:06.780","Text":"Those are that the logarithm of"},{"Start":"10:06.780 ","End":"10:13.265","Text":"something like a^n is n times natural log of a."},{"Start":"10:13.265 ","End":"10:16.500","Text":"This works for any log of any base."},{"Start":"10:16.720 ","End":"10:26.525","Text":"The natural log of a product ab is the log of a plus the log of b."},{"Start":"10:26.525 ","End":"10:30.500","Text":"Of course, these equalities work the other way around too."},{"Start":"10:30.500 ","End":"10:33.590","Text":"What I mean is if I\u0027m given this expression,"},{"Start":"10:33.590 ","End":"10:34.969","Text":"I can get this expression."},{"Start":"10:34.969 ","End":"10:44.555","Text":"The reason I\u0027d say that is that I can rewrite this as the natural log of x squared,"},{"Start":"10:44.555 ","End":"10:49.655","Text":"and I can write this as the natural log of y^4."},{"Start":"10:49.655 ","End":"10:52.445","Text":"Let me get organized here."},{"Start":"10:52.445 ","End":"10:55.395","Text":"Limit x goes to 1,"},{"Start":"10:55.395 ","End":"11:04.060","Text":"y goes to 1 over x^4 y^8 minus 1."},{"Start":"11:05.360 ","End":"11:12.045","Text":"Okay, let\u0027s take it a step further using these logarithm rules."},{"Start":"11:12.045 ","End":"11:15.125","Text":"I can use the other rule."},{"Start":"11:15.125 ","End":"11:18.895","Text":"I have here, natural log of something plus natural log of something."},{"Start":"11:18.895 ","End":"11:26.770","Text":"I can write this as the limit of the natural log of x"},{"Start":"11:26.770 ","End":"11:35.075","Text":"squared times y^4 over x^4 y^8 minus 1."},{"Start":"11:35.075 ","End":"11:38.380","Text":"Now, do you see what we\u0027re going to substitute?"},{"Start":"11:38.380 ","End":"11:43.865","Text":"This thing is exactly the square of this thing."},{"Start":"11:43.865 ","End":"11:47.210","Text":"Again, there\u0027s rules of logarithms we could apply here."},{"Start":"11:47.210 ","End":"11:48.590","Text":"But if we take this squared,"},{"Start":"11:48.590 ","End":"11:51.170","Text":"we square each 1 separately and then we get this."},{"Start":"11:51.170 ","End":"12:00.615","Text":"If I take t is equal to x squared y^4,"},{"Start":"12:00.615 ","End":"12:06.465","Text":"then I will get the limit of"},{"Start":"12:06.465 ","End":"12:15.735","Text":"natural log of t over t squared minus 1."},{"Start":"12:15.735 ","End":"12:17.970","Text":"Now, where does t go to?"},{"Start":"12:17.970 ","End":"12:20.760","Text":"Well, x goes to 1 and y goes to 1,"},{"Start":"12:20.760 ","End":"12:23.700","Text":"so clearly, t goes to 1."},{"Start":"12:23.700 ","End":"12:27.030","Text":"If we just substitute,"},{"Start":"12:27.030 ","End":"12:31.960","Text":"then we\u0027ll get, again, 0 over 0."},{"Start":"12:32.810 ","End":"12:36.515","Text":"We\u0027ll use L\u0027Hopital\u0027s rule again."},{"Start":"12:36.515 ","End":"12:41.259","Text":"Differentiate top and bottom and get the limit."},{"Start":"12:41.259 ","End":"12:43.915","Text":"t still goes to 1."},{"Start":"12:43.915 ","End":"12:50.595","Text":"The derivative of this is 1 over t. The derivative of this is 2t."},{"Start":"12:50.595 ","End":"12:54.500","Text":"At this point there\u0027s no problem putting t equals 1."},{"Start":"12:54.500 ","End":"12:58.295","Text":"Clearly, the answer is 1/2,"},{"Start":"12:58.295 ","End":"13:00.140","Text":"and that is the answer."},{"Start":"13:00.140 ","End":"13:05.405","Text":"As you see, the techniques of substitution,"},{"Start":"13:05.405 ","End":"13:08.645","Text":"sometimes it\u0027s easy to see what the substitution is,"},{"Start":"13:08.645 ","End":"13:13.610","Text":"but sometimes it\u0027s not immediately clear and you have to work a bit at it."},{"Start":"13:13.610 ","End":"13:20.150","Text":"There are plenty more examples in the exercises following the tutorials,"},{"Start":"13:20.150 ","End":"13:23.135","Text":"and you\u0027re welcome to try them."},{"Start":"13:23.135 ","End":"13:26.570","Text":"Meanwhile, we\u0027ve crossed off technique number"},{"Start":"13:26.570 ","End":"13:30.065","Text":"1 from the list of 4 techniques we\u0027re going to learn."},{"Start":"13:30.065 ","End":"13:33.420","Text":"In the next clip, we\u0027ll do the next technique."}],"ID":8882},{"Watched":false,"Name":"Polar Substitution","Duration":"10m 15s","ChapterTopicVideoID":8509,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.770","Text":"Next we come to technique number 2 for finding the limit of a function of 2 variables,"},{"Start":"00:05.770 ","End":"00:08.890","Text":"and this one is called polar substitution."},{"Start":"00:08.890 ","End":"00:11.425","Text":"I\u0027ll start with an example."},{"Start":"00:11.425 ","End":"00:19.870","Text":"Let\u0027s take the limit as x,y goes"},{"Start":"00:19.870 ","End":"00:25.180","Text":"to 0,0 of x squared"},{"Start":"00:25.180 ","End":"00:31.910","Text":"y over x squared plus y squared."},{"Start":"00:32.130 ","End":"00:35.050","Text":"One of the things that indicates to me"},{"Start":"00:35.050 ","End":"00:39.590","Text":"polar substitution is the presence of the x squared plus y squared."},{"Start":"00:39.590 ","End":"00:42.875","Text":"I\u0027m going to show you what the substitution is,"},{"Start":"00:42.875 ","End":"00:46.870","Text":"but don\u0027t worry if you don\u0027t know where it came from or what it means."},{"Start":"00:46.870 ","End":"00:49.330","Text":"The polar substitution is as follow,"},{"Start":"00:49.330 ","End":"00:55.200","Text":"we set x equals r cosine Theta."},{"Start":"00:55.200 ","End":"00:58.500","Text":"Don\u0027t be frightened by the letter Theta, it could be Alpha,"},{"Start":"00:58.500 ","End":"01:00.540","Text":"it\u0027s just customary to write Theta,"},{"Start":"01:00.540 ","End":"01:06.105","Text":"and y equals r times sine Theta."},{"Start":"01:06.105 ","End":"01:09.910","Text":"If you compute x squared plus y squared,"},{"Start":"01:09.910 ","End":"01:12.535","Text":"because of trigonometrical identities,"},{"Start":"01:12.535 ","End":"01:16.730","Text":"we\u0027ll find that this is equal to r squared,"},{"Start":"01:16.730 ","End":"01:20.310","Text":"and you would replace the limit,"},{"Start":"01:20.310 ","End":"01:25.420","Text":"instead of x,y goes to 0,0,"},{"Start":"01:25.420 ","End":"01:30.370","Text":"you replace that with r goes to 0,"},{"Start":"01:30.370 ","End":"01:33.030","Text":"but from the positive direction."},{"Start":"01:33.030 ","End":"01:37.525","Text":"r is considered to be a non-negative variable."},{"Start":"01:37.525 ","End":"01:40.370","Text":"I\u0027m just presenting it to you technically,"},{"Start":"01:40.370 ","End":"01:43.280","Text":"although there is a logic behind all this and a meaning."},{"Start":"01:43.280 ","End":"01:46.055","Text":"In this case, if we make the substitution,"},{"Start":"01:46.055 ","End":"01:51.505","Text":"we get the limit r goes to 0 plus,"},{"Start":"01:51.505 ","End":"01:53.300","Text":"approaches 0 from the right."},{"Start":"01:53.300 ","End":"01:57.079","Text":"Now the x squared plus y squared is r squared,"},{"Start":"01:57.079 ","End":"02:02.160","Text":"x squared is r squared cosine squared Theta,"},{"Start":"02:03.140 ","End":"02:08.050","Text":"and y is r sine Theta."},{"Start":"02:08.360 ","End":"02:11.805","Text":"We can simplify this a bit."},{"Start":"02:11.805 ","End":"02:19.040","Text":"This is equal to the limit as r goes to 0 plus"},{"Start":"02:19.040 ","End":"02:27.595","Text":"of r cosine squared Theta sine Theta."},{"Start":"02:27.595 ","End":"02:29.960","Text":"Another important thing to remember,"},{"Start":"02:29.960 ","End":"02:32.150","Text":"especially in this kind of exercise,"},{"Start":"02:32.150 ","End":"02:37.080","Text":"is that the sine and the cosine are bounded."},{"Start":"02:37.080 ","End":"02:42.255","Text":"The absolute value of sine Theta is less than or equal to 1,"},{"Start":"02:42.255 ","End":"02:47.470","Text":"and also the absolute value of cosine Theta is less than or equal to 1."},{"Start":"02:47.470 ","End":"02:52.370","Text":"This is because each of them is between minus 1 and 1."},{"Start":"02:52.370 ","End":"02:55.460","Text":"The sine and the cosine oscillate,"},{"Start":"02:55.460 ","End":"02:57.285","Text":"but always between 1 and minus 1,"},{"Start":"02:57.285 ","End":"03:00.815","Text":"which means that their absolute size is never greater than 1."},{"Start":"03:00.815 ","End":"03:02.795","Text":"I\u0027m going to use that here."},{"Start":"03:02.795 ","End":"03:04.725","Text":"Now back to this limit here,"},{"Start":"03:04.725 ","End":"03:07.640","Text":"there\u0027s 2 parts, there\u0027s the r and then there\u0027s the rest of it."},{"Start":"03:07.640 ","End":"03:11.190","Text":"Now r goes to 0,"},{"Start":"03:11.530 ","End":"03:17.010","Text":"and this thing is bounded."},{"Start":"03:17.170 ","End":"03:22.715","Text":"It\u0027s bounded because the absolute value of"},{"Start":"03:22.715 ","End":"03:30.080","Text":"cosine squared Theta sine Theta is also going to be less than or equal to,"},{"Start":"03:30.080 ","End":"03:31.910","Text":"each of these is less than or equal to 1,"},{"Start":"03:31.910 ","End":"03:34.205","Text":"so it\u0027s also less than or equal to 1,"},{"Start":"03:34.205 ","End":"03:36.740","Text":"which means it\u0027s between minus 1 and 1."},{"Start":"03:36.740 ","End":"03:40.730","Text":"Now, it\u0027s well known that something that goes to"},{"Start":"03:40.730 ","End":"03:45.425","Text":"0 times something bounded also tends to 0."},{"Start":"03:45.425 ","End":"03:51.245","Text":"This is therefore equal to 0, and that\u0027s the answer."},{"Start":"03:51.245 ","End":"03:55.175","Text":"If you had tried to do this with technique number 1 substitution,"},{"Start":"03:55.175 ","End":"03:57.560","Text":"there\u0027s no way you would find anything meaningful to"},{"Start":"03:57.560 ","End":"04:00.830","Text":"substitute t for it, it just wouldn\u0027t work."},{"Start":"04:00.830 ","End":"04:05.075","Text":"This is where technique number 2 comes to our aid."},{"Start":"04:05.075 ","End":"04:08.660","Text":"Let\u0027s go for another example."},{"Start":"04:08.660 ","End":"04:19.330","Text":"This time I\u0027ll take the limit as"},{"Start":"04:19.330 ","End":"04:28.160","Text":"x,y goes to 0,0 of 4x squared minus"},{"Start":"04:28.160 ","End":"04:33.035","Text":"3xy squared plus 4y"},{"Start":"04:33.035 ","End":"04:38.375","Text":"squared over x squared plus y squared."},{"Start":"04:38.375 ","End":"04:45.755","Text":"Once again, the x squared plus y squared is often a hint to use the polar substitution,"},{"Start":"04:45.755 ","End":"04:50.060","Text":"and they\u0027ll remind you of what the polar substitution is."},{"Start":"04:50.060 ","End":"04:54.530","Text":"Well, I just copied from above what we had."},{"Start":"04:55.840 ","End":"05:05.135","Text":"We get the limit as r goes to 0 plus,"},{"Start":"05:05.135 ","End":"05:09.450","Text":"denominator is easy to substitute."},{"Start":"05:09.830 ","End":"05:14.460","Text":"x squared plus y squared is r squared."},{"Start":"05:14.460 ","End":"05:19.280","Text":"The numerator, if you look at it,"},{"Start":"05:19.280 ","End":"05:23.600","Text":"what I have in the numerator is if I just look at the first and last terms,"},{"Start":"05:23.600 ","End":"05:28.245","Text":"I\u0027ve got 4x squared plus 4y squared,"},{"Start":"05:28.245 ","End":"05:32.265","Text":"which equals 4x squared plus y squared,"},{"Start":"05:32.265 ","End":"05:37.690","Text":"which equals 4r squared."},{"Start":"05:38.240 ","End":"05:47.935","Text":"Here I have 4r squared and from here I get minus 3."},{"Start":"05:47.935 ","End":"05:53.700","Text":"Now x is r cosine Theta and y is r sine Theta."},{"Start":"05:53.700 ","End":"05:56.749","Text":"If I take this times this squared,"},{"Start":"05:56.749 ","End":"06:03.300","Text":"I\u0027ll get r times r times r. I\u0027ll collect all those together and I\u0027ll get cosine Theta,"},{"Start":"06:03.300 ","End":"06:05.220","Text":"sine Theta, sine Theta,"},{"Start":"06:05.220 ","End":"06:11.440","Text":"so cosine Theta sine squared Theta."},{"Start":"06:12.350 ","End":"06:15.825","Text":"The limit r goes to 0,"},{"Start":"06:15.825 ","End":"06:21.300","Text":"and this equals, I can split it up into 2."},{"Start":"06:21.300 ","End":"06:25.770","Text":"It\u0027s the limit as r goes to 0 plus,"},{"Start":"06:25.770 ","End":"06:28.750","Text":"this over this is just 4."},{"Start":"06:28.900 ","End":"06:33.290","Text":"If I take this over r squared,"},{"Start":"06:33.290 ","End":"06:43.930","Text":"I\u0027ll get minus 3r times cosine Theta sine squared Theta."},{"Start":"06:43.930 ","End":"06:48.185","Text":"Now just as before in the previous example,"},{"Start":"06:48.185 ","End":"06:51.005","Text":"we can do the same thing here."},{"Start":"06:51.005 ","End":"06:57.445","Text":"This is bounded, and this goes to 0,"},{"Start":"06:57.445 ","End":"07:03.230","Text":"so altogether, 0 times bounded is 0."},{"Start":"07:03.230 ","End":"07:04.280","Text":"The 3 doesn\u0027t matter,"},{"Start":"07:04.280 ","End":"07:05.975","Text":"3 times 0 is 0,"},{"Start":"07:05.975 ","End":"07:11.285","Text":"so basically this limit is equal to 4,"},{"Start":"07:11.285 ","End":"07:13.835","Text":"and that\u0027s it for this example."},{"Start":"07:13.835 ","End":"07:17.240","Text":"The last example here is different."},{"Start":"07:17.240 ","End":"07:21.485","Text":"This example is going to show us how this technique of"},{"Start":"07:21.485 ","End":"07:26.765","Text":"polar substitution can tell us that something doesn\u0027t have a limit."},{"Start":"07:26.765 ","End":"07:28.474","Text":"Let\u0027s look at the following."},{"Start":"07:28.474 ","End":"07:37.475","Text":"The limit as x,y goes to 0,0 of y over x."},{"Start":"07:37.475 ","End":"07:40.715","Text":"If we make the same substitution,"},{"Start":"07:40.715 ","End":"07:45.965","Text":"and here I\u0027ve copied it again because you may not have memorized it."},{"Start":"07:45.965 ","End":"07:50.240","Text":"Making the substitution, we get that this is equal"},{"Start":"07:50.240 ","End":"08:00.005","Text":"to the limit as r goes to 0 plus,"},{"Start":"08:00.005 ","End":"08:10.120","Text":"y is r sine Theta and x is r cosine Theta."},{"Start":"08:10.120 ","End":"08:17.485","Text":"What we\u0027re left with is that r cancels and this is the limit r"},{"Start":"08:17.485 ","End":"08:24.290","Text":"goes to 0 of sine over cosine is tangent of tangent Theta."},{"Start":"08:24.290 ","End":"08:28.415","Text":"Now, whenever you\u0027re left with an expression that just involves Theta,"},{"Start":"08:28.415 ","End":"08:33.050","Text":"it doesn\u0027t have a limit because r can go to 0."},{"Start":"08:33.050 ","End":"08:36.680","Text":"When r goes to 0, the limit is just this thing here,"},{"Start":"08:36.680 ","End":"08:38.930","Text":"it\u0027s a constant as far as r goes."},{"Start":"08:38.930 ","End":"08:40.595","Text":"It\u0027s just tangent of Theta,"},{"Start":"08:40.595 ","End":"08:43.075","Text":"but Theta could be anything,"},{"Start":"08:43.075 ","End":"08:45.230","Text":"so it depends on Theta."},{"Start":"08:45.230 ","End":"08:49.955","Text":"In any event, you just have to remember that when you make the substitution,"},{"Start":"08:49.955 ","End":"08:53.450","Text":"r disappears and we\u0027re dependent only on Theta,"},{"Start":"08:53.450 ","End":"08:55.910","Text":"then we don\u0027t have a limit."},{"Start":"08:55.910 ","End":"08:59.915","Text":"We can say that there is no limit."},{"Start":"08:59.915 ","End":"09:07.560","Text":"This contrasts with the previous example where we had Theta in the limit,"},{"Start":"09:07.560 ","End":"09:09.285","Text":"but we still had r in it."},{"Start":"09:09.285 ","End":"09:14.140","Text":"Now, although we didn\u0027t know what Theta was and Theta could be anything, nevertheless,"},{"Start":"09:14.140 ","End":"09:17.300","Text":"it was bounded and multiplied by something going to 0,"},{"Start":"09:17.300 ","End":"09:19.985","Text":"so it didn\u0027t matter what Theta was."},{"Start":"09:19.985 ","End":"09:24.880","Text":"In any event, we got the answer here 4,"},{"Start":"09:24.880 ","End":"09:27.680","Text":"but here it very much matters what Theta is."},{"Start":"09:27.680 ","End":"09:30.770","Text":"There\u0027s no way it\u0027s going to disappear or not make a difference."},{"Start":"09:30.770 ","End":"09:33.740","Text":"As I said, if you get just an expression in Theta,"},{"Start":"09:33.740 ","End":"09:35.190","Text":"the answer is that there is no limit,"},{"Start":"09:35.190 ","End":"09:37.415","Text":"the original limit does not exist."},{"Start":"09:37.415 ","End":"09:40.580","Text":"I\u0027m basically done with this section,"},{"Start":"09:40.580 ","End":"09:43.130","Text":"but I\u0027d like to mention that not everyone,"},{"Start":"09:43.130 ","End":"09:47.120","Text":"not in every college do they teach this technique."},{"Start":"09:47.120 ","End":"09:52.505","Text":"Instead, they sometimes use something called the Sandwich Theorem,"},{"Start":"09:52.505 ","End":"09:55.550","Text":"which requires you to be very creative,"},{"Start":"09:55.550 ","End":"10:00.590","Text":"let\u0027s say, whereas here it\u0027s pretty much technical and I prefer this substitution."},{"Start":"10:00.590 ","End":"10:05.180","Text":"Besides, you\u0027re going to learn this substitution any way later on."},{"Start":"10:05.180 ","End":"10:09.890","Text":"For example, in the section on multiple integrals,"},{"Start":"10:09.890 ","End":"10:16.010","Text":"integrals of functions of 2 variables and so on. On to the next."}],"ID":8883},{"Watched":false,"Name":"Iterated Limits","Duration":"7m 11s","ChapterTopicVideoID":8510,"CourseChapterTopicPlaylistID":4970,"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":"Continuing with the limit of a function of 2 variables,"},{"Start":"00:04.710 ","End":"00:07.230","Text":"we now come to technique number 3,"},{"Start":"00:07.230 ","End":"00:09.840","Text":"which is called iterated limits."},{"Start":"00:09.840 ","End":"00:13.500","Text":"From the start, I want to stress that this technique can only be"},{"Start":"00:13.500 ","End":"00:17.085","Text":"used for non-existence of a limit."},{"Start":"00:17.085 ","End":"00:21.800","Text":"I want to prove that something has a limit or find its limit, this is no good."},{"Start":"00:21.800 ","End":"00:26.180","Text":"This is only useful for proving nonexistence and it sometimes works, sometimes, doesn\u0027t."},{"Start":"00:26.180 ","End":"00:27.984","Text":"So you need to come to the conclusion,"},{"Start":"00:27.984 ","End":"00:30.545","Text":"limit doesn\u0027t exist or don\u0027t know."},{"Start":"00:30.545 ","End":"00:31.790","Text":"I\u0027ll just write that down."},{"Start":"00:31.790 ","End":"00:35.040","Text":"It\u0027s for non-existence."},{"Start":"00:35.050 ","End":"00:39.860","Text":"Now, what is this iterated limits concept?"},{"Start":"00:39.860 ","End":"00:42.050","Text":"Normally when we take a limit,"},{"Start":"00:42.050 ","End":"00:44.450","Text":"we look at it as the limit,"},{"Start":"00:44.450 ","End":"00:47.765","Text":"say, x,y goes to,"},{"Start":"00:47.765 ","End":"00:49.820","Text":"let\u0027s take an example."},{"Start":"00:49.820 ","End":"00:52.265","Text":"Actually a real example,"},{"Start":"00:52.265 ","End":"01:01.535","Text":"x,y goes to 0,0 of 4x plus 10y over x plus y."},{"Start":"01:01.535 ","End":"01:04.820","Text":"Now, this is called just the limit,"},{"Start":"01:04.820 ","End":"01:09.215","Text":"and sometimes it\u0027s called the double limit because it\u0027s in 2 variables."},{"Start":"01:09.215 ","End":"01:13.610","Text":"There\u0027s another concept which is called the iterated limit,"},{"Start":"01:13.610 ","End":"01:17.480","Text":"which is not to take the limit all at once as x,"},{"Start":"01:17.480 ","End":"01:18.860","Text":"y goes to 0, 0,"},{"Start":"01:18.860 ","End":"01:20.375","Text":"but to take separately,"},{"Start":"01:20.375 ","End":"01:23.915","Text":"the x goes to 0 and the y goes to 0."},{"Start":"01:23.915 ","End":"01:27.090","Text":"Now, there\u0027s 2 ways of doing this."},{"Start":"01:29.210 ","End":"01:35.530","Text":"Well, I\u0027m working from inside out."},{"Start":"01:35.530 ","End":"01:45.805","Text":"We can take first the limit as x goes to 0 of 4x plus 10y over x plus y."},{"Start":"01:45.805 ","End":"01:47.665","Text":"When we take x to 0,"},{"Start":"01:47.665 ","End":"01:50.305","Text":"we treat y as if it was a constant."},{"Start":"01:50.305 ","End":"01:52.480","Text":"When we finish taking x to 0,"},{"Start":"01:52.480 ","End":"01:54.400","Text":"we\u0027ll have a function of y."},{"Start":"01:54.400 ","End":"02:00.055","Text":"Then we can take the limit as y goes to 0 of the answer."},{"Start":"02:00.055 ","End":"02:01.645","Text":"On the other hand,"},{"Start":"02:01.645 ","End":"02:03.175","Text":"we could do it the other way."},{"Start":"02:03.175 ","End":"02:13.725","Text":"We could, first of all take the limit as y goes to 0 of 4x plus 10y over x plus y."},{"Start":"02:13.725 ","End":"02:20.030","Text":"Then we treat x like a constant and we take y to 0 and we\u0027re left with a function of x."},{"Start":"02:20.030 ","End":"02:24.655","Text":"Then we can take the limit as x goes to 0 of that."},{"Start":"02:24.655 ","End":"02:29.630","Text":"The technique of iterated limits somehow ties in the non-existence or existence"},{"Start":"02:29.630 ","End":"02:34.910","Text":"of this with the existence of these 2 and whether they\u0027re equal or not."},{"Start":"02:34.910 ","End":"02:41.075","Text":"That\u0027s in general. First, why don\u0027t we just compute these 2 iterated limits?"},{"Start":"02:41.075 ","End":"02:43.010","Text":"Let\u0027s do the first 1."},{"Start":"02:43.010 ","End":"02:45.680","Text":"When x goes to 0,"},{"Start":"02:45.680 ","End":"02:48.890","Text":"we can just substitute x equals 0."},{"Start":"02:48.890 ","End":"02:53.195","Text":"This disappears and this disappears and we get"},{"Start":"02:53.195 ","End":"03:03.875","Text":"the limit as y goes to 0 of 10y over y and this is equal to 10."},{"Start":"03:03.875 ","End":"03:07.520","Text":"On the other hand, if we do it the other way, if y goes to 0,"},{"Start":"03:07.520 ","End":"03:13.880","Text":"we can just substitute it and get the limit of 4x over x. I\u0027m taking"},{"Start":"03:13.880 ","End":"03:21.070","Text":"the limit when x goes to 0 here and that\u0027s equal to 4."},{"Start":"03:21.070 ","End":"03:27.365","Text":"What I have here is that these limits both exist,"},{"Start":"03:27.365 ","End":"03:29.735","Text":"but they\u0027re not equal."},{"Start":"03:29.735 ","End":"03:32.735","Text":"If such a thing happens,"},{"Start":"03:32.735 ","End":"03:36.605","Text":"then this does not have a limit."},{"Start":"03:36.605 ","End":"03:40.450","Text":"That\u0027s what the claim is."},{"Start":"03:40.450 ","End":"03:45.430","Text":"If you take the 2 iterated limits and they exist but not equal,"},{"Start":"03:45.430 ","End":"03:47.470","Text":"this doesn\u0027t have a limit."},{"Start":"03:47.470 ","End":"03:49.870","Text":"Let me write down the theorem."},{"Start":"03:49.870 ","End":"03:53.530","Text":"If the 2 iterated limits exist,"},{"Start":"03:53.530 ","End":"03:57.040","Text":"meaning they\u0027re finite, but are not equal,"},{"Start":"03:57.040 ","End":"04:01.180","Text":"as in our case, then the double limit does not exist."},{"Start":"04:01.180 ","End":"04:03.115","Text":"Because of this theorem,"},{"Start":"04:03.115 ","End":"04:08.470","Text":"we can say that this does not exist, follows from this."},{"Start":"04:08.470 ","End":"04:11.800","Text":"Suppose we got the first limit was also equal to 4,"},{"Start":"04:11.800 ","End":"04:13.835","Text":"what can we conclude then?"},{"Start":"04:13.835 ","End":"04:15.435","Text":"The answer is nothing."},{"Start":"04:15.435 ","End":"04:17.755","Text":"If these 2 come out to be equal,"},{"Start":"04:17.755 ","End":"04:21.175","Text":"the original double limit might or might not exist."},{"Start":"04:21.175 ","End":"04:25.355","Text":"I think I\u0027ll write that warning because people fall into that trap."},{"Start":"04:25.355 ","End":"04:29.320","Text":"Let me write a warning."},{"Start":"04:29.610 ","End":"04:32.260","Text":"I just wrote down what I said,"},{"Start":"04:32.260 ","End":"04:33.880","Text":"that if the 2 limits are equal,"},{"Start":"04:33.880 ","End":"04:38.505","Text":"we still can\u0027t conclude that the double limit exists or doesn\u0027t exist."},{"Start":"04:38.505 ","End":"04:43.285","Text":"This is only useful for proving that the double limit does not exist."},{"Start":"04:43.285 ","End":"04:46.060","Text":"Let\u0027s go on to another example."},{"Start":"04:46.060 ","End":"04:50.959","Text":"Let\u0027s take the limit."},{"Start":"04:51.120 ","End":"05:00.280","Text":"Again as x,y goes to 0,0"},{"Start":"05:00.280 ","End":"05:09.880","Text":"of x squared plus y squared over y plus x squared."},{"Start":"05:10.370 ","End":"05:13.890","Text":"Let\u0027s compute the 2 iterated limits."},{"Start":"05:13.890 ","End":"05:17.205","Text":"First, we take limit, let\u0027s say,"},{"Start":"05:17.205 ","End":"05:22.260","Text":"x goes to 0 of limit y goes to"},{"Start":"05:22.260 ","End":"05:30.425","Text":"0 of x squared plus y squared over y plus x squared."},{"Start":"05:30.425 ","End":"05:40.370","Text":"Then we\u0027ll compute the limit as y goes to 0 of the limit as x goes to 0 of same thing,"},{"Start":"05:40.370 ","End":"05:46.229","Text":"x squared plus y squared over y plus x squared."},{"Start":"05:46.350 ","End":"05:50.705","Text":"Notice that these are usually very easy to compute."},{"Start":"05:50.705 ","End":"05:53.710","Text":"There\u0027s not a lot of work in this."},{"Start":"05:54.080 ","End":"05:57.044","Text":"Limit y goes to 0,"},{"Start":"05:57.044 ","End":"05:59.130","Text":"so this is 0, this is 0."},{"Start":"05:59.130 ","End":"06:02.320","Text":"We get x squared over x squared,"},{"Start":"06:02.690 ","End":"06:07.455","Text":"which is the limit as x goes to 0."},{"Start":"06:07.455 ","End":"06:09.915","Text":"So x squared over x squared."},{"Start":"06:09.915 ","End":"06:11.970","Text":"X squared over x squared is 1,"},{"Start":"06:11.970 ","End":"06:13.740","Text":"so this is equal to 1."},{"Start":"06:13.740 ","End":"06:19.150","Text":"On the other hand, this limit is the limit as y goes to 0."},{"Start":"06:19.150 ","End":"06:21.430","Text":"If we put x equals 0 here,"},{"Start":"06:21.430 ","End":"06:24.910","Text":"the x squared disappears."},{"Start":"06:24.910 ","End":"06:27.620","Text":"We have y squared over y."},{"Start":"06:28.080 ","End":"06:30.260","Text":"1 of the y\u0027s cancels,"},{"Start":"06:30.260 ","End":"06:33.785","Text":"this just equals y and the limit is 0."},{"Start":"06:33.785 ","End":"06:37.475","Text":"These 2 are not equal."},{"Start":"06:37.475 ","End":"06:44.730","Text":"This limit is non-existent."},{"Start":"06:44.770 ","End":"06:48.005","Text":"This goes very easily and in fact,"},{"Start":"06:48.005 ","End":"06:53.780","Text":"I recommend almost any exercise you get with the limits to do"},{"Start":"06:53.780 ","End":"07:01.325","Text":"this iterated limit technique first because it\u0027s just so quick and then go to the others."},{"Start":"07:01.325 ","End":"07:02.810","Text":"I\u0027m basically done here."},{"Start":"07:02.810 ","End":"07:06.740","Text":"I just want to remind you that there are exercises for practice following"},{"Start":"07:06.740 ","End":"07:12.660","Text":"the tutorials and just try out some exercises. That\u0027s it."}],"ID":8884},{"Watched":false,"Name":"Limit along a Path","Duration":"13m 35s","ChapterTopicVideoID":8511,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:07.170","Text":"Now we come to technique Number 4 regarding the limit of a function of 2 variables."},{"Start":"00:07.170 ","End":"00:10.635","Text":"Like the previous 1 Technique Number 3,"},{"Start":"00:10.635 ","End":"00:14.955","Text":"this is a technique used only to prove nonexistence."},{"Start":"00:14.955 ","End":"00:17.595","Text":"I call it limit along a path."},{"Start":"00:17.595 ","End":"00:19.905","Text":"That\u0027s just a name I chose."},{"Start":"00:19.905 ","End":"00:22.230","Text":"I\u0027ll show you what it is in a minute,"},{"Start":"00:22.230 ","End":"00:26.280","Text":"but let\u0027s first of all start right away with an example."},{"Start":"00:26.280 ","End":"00:30.960","Text":"Let\u0027s take the limit as x,"},{"Start":"00:30.960 ","End":"00:33.240","Text":"y goes to 0,"},{"Start":"00:33.240 ","End":"00:42.475","Text":"0 of xy over x squared plus y squared."},{"Start":"00:42.475 ","End":"00:48.765","Text":"First, I\u0027d like to try it with the iterated limit method."},{"Start":"00:48.765 ","End":"00:51.470","Text":"We\u0027ll take the 2 limits."},{"Start":"00:51.470 ","End":"00:54.290","Text":"First of all, x goes to 0,"},{"Start":"00:54.290 ","End":"00:56.645","Text":"y goes to 0,"},{"Start":"00:56.645 ","End":"01:01.030","Text":"xy over x squared plus y squared."},{"Start":"01:01.030 ","End":"01:03.765","Text":"Here, y goes to 0."},{"Start":"01:03.765 ","End":"01:06.980","Text":"I\u0027ve got 0 over x squared,"},{"Start":"01:06.980 ","End":"01:10.445","Text":"which is 0, and this is 0."},{"Start":"01:10.445 ","End":"01:15.124","Text":"The other way, limit y goes to 0,"},{"Start":"01:15.124 ","End":"01:17.480","Text":"limit x goes to 0,"},{"Start":"01:17.480 ","End":"01:21.020","Text":"xy over x squared plus y squared."},{"Start":"01:21.020 ","End":"01:23.615","Text":"Same thing, but let x go to 0."},{"Start":"01:23.615 ","End":"01:25.930","Text":"Then here I get 0,"},{"Start":"01:25.930 ","End":"01:31.850","Text":"and this is non-zero because it\u0027s y squared in the denominator, so it\u0027s 0."},{"Start":"01:31.850 ","End":"01:35.075","Text":"I\u0027ve got limit that x goes to 0 of 0 again,"},{"Start":"01:35.075 ","End":"01:39.240","Text":"and it\u0027s 0, and these 2 are equal."},{"Start":"01:39.310 ","End":"01:42.080","Text":"However, when they\u0027re equal,"},{"Start":"01:42.080 ","End":"01:44.495","Text":"we cannot conclude this."},{"Start":"01:44.495 ","End":"01:47.135","Text":"Does it have a limit or doesn\u0027t it?"},{"Start":"01:47.135 ","End":"01:49.140","Text":"This technique didn\u0027t work for us,"},{"Start":"01:49.140 ","End":"01:51.290","Text":"so I\u0027m going to introduce another technique."},{"Start":"01:51.290 ","End":"01:53.945","Text":"Let me just give you a bit background."},{"Start":"01:53.945 ","End":"01:55.655","Text":"When I take a limit,"},{"Start":"01:55.655 ","End":"02:00.740","Text":"a double limit, there\u0027s a concept called taking a limit along the path."},{"Start":"02:00.740 ","End":"02:05.190","Text":"Instead of just going to 0,0 together,"},{"Start":"02:05.190 ","End":"02:08.195","Text":"we take it to 0,0 along a specific path."},{"Start":"02:08.195 ","End":"02:13.950","Text":"For example, if I take the function y equals x squared,"},{"Start":"02:14.750 ","End":"02:21.305","Text":"I can say that I\u0027m going to 0,0 along this path."},{"Start":"02:21.305 ","End":"02:25.820","Text":"Another path example would be to take y equals"},{"Start":"02:25.820 ","End":"02:32.420","Text":"x and I could go to 0 along this path."},{"Start":"02:32.420 ","End":"02:41.380","Text":"Still another example, the function y equals 2x and go to 0, 0 along this."},{"Start":"02:41.380 ","End":"02:43.505","Text":"I forgot to label these."},{"Start":"02:43.505 ","End":"02:45.150","Text":"There\u0027s actually a theorem,"},{"Start":"02:45.150 ","End":"02:48.970","Text":"I\u0027m not going to write it down because you don\u0027t need to use it directly,"},{"Start":"02:48.970 ","End":"02:52.465","Text":"but if the function has a limit,"},{"Start":"02:52.465 ","End":"02:55.390","Text":"say as x, y goes to 0,0,"},{"Start":"02:55.390 ","End":"03:00.310","Text":"then if I take any path and I let xy go to 00 along this path,"},{"Start":"03:00.310 ","End":"03:04.105","Text":"they all exist and they\u0027re all equal to this limit."},{"Start":"03:04.105 ","End":"03:06.100","Text":"The idea is to find"},{"Start":"03:06.100 ","End":"03:12.144","Text":"2 different paths and I go to 0,0 along each of them and I get a different result."},{"Start":"03:12.144 ","End":"03:13.465","Text":"That\u0027s the idea."},{"Start":"03:13.465 ","End":"03:17.460","Text":"Now let me show you in more detail with an example."},{"Start":"03:17.460 ","End":"03:21.410","Text":"First, let\u0027s take the limit along 1 path,"},{"Start":"03:21.410 ","End":"03:25.220","Text":"and I\u0027ll choose this green path, y equals x."},{"Start":"03:25.220 ","End":"03:31.490","Text":"The way we do that is we take this limit and just substitute y equals x."},{"Start":"03:31.490 ","End":"03:37.650","Text":"We get the limit as x goes to 0,"},{"Start":"03:37.650 ","End":"03:39.335","Text":"and if y equals x,"},{"Start":"03:39.335 ","End":"03:46.625","Text":"we get x squared over x squared plus x squared."},{"Start":"03:46.625 ","End":"03:50.570","Text":"Notice that the path I chose this y equals x,"},{"Start":"03:50.570 ","End":"03:54.785","Text":"contains the point 0,0."},{"Start":"03:54.785 ","End":"04:00.890","Text":"In other words, it goes through the origin in this case because of the 0,0."},{"Start":"04:00.890 ","End":"04:11.369","Text":"What we get is the limit as x goes to 0 of x squared over 2x squared,"},{"Start":"04:11.369 ","End":"04:14.010","Text":"and the x squared cancel, it\u0027s a half,"},{"Start":"04:14.010 ","End":"04:16.730","Text":"the limit is just 1/2."},{"Start":"04:16.730 ","End":"04:18.605","Text":"On the other hand,"},{"Start":"04:18.605 ","End":"04:23.455","Text":"if I take the limit alongside the red 1,"},{"Start":"04:23.455 ","End":"04:25.805","Text":"y equals to x,"},{"Start":"04:25.805 ","End":"04:30.530","Text":"then what I get is I substitute y equals to x,"},{"Start":"04:30.530 ","End":"04:32.150","Text":"x goes to 0."},{"Start":"04:32.150 ","End":"04:35.130","Text":"That\u0027s the 0 from here."},{"Start":"04:35.930 ","End":"04:40.700","Text":"Let me just scroll so we can see the original."},{"Start":"04:40.700 ","End":"04:45.205","Text":"It\u0027s x times y, so it\u0027s 2x squared, sorry."},{"Start":"04:45.205 ","End":"04:52.965","Text":"Y is 2x over x squared plus 2x squared is 4x squared."},{"Start":"04:52.965 ","End":"04:58.340","Text":"This equals the limit of 2x squared over 5 x squared."},{"Start":"04:58.340 ","End":"05:00.275","Text":"The x squared cancels,"},{"Start":"05:00.275 ","End":"05:04.220","Text":"and the limit of a constant 2/5 is just 2/5."},{"Start":"05:04.220 ","End":"05:09.150","Text":"Now, these 2 are definitely not equal,"},{"Start":"05:09.150 ","End":"05:12.635","Text":"1/2 is not equal to 2/5."},{"Start":"05:12.635 ","End":"05:15.510","Text":"As required, we found 2 different paths,"},{"Start":"05:15.510 ","End":"05:16.710","Text":"the green 1 and the red 1,"},{"Start":"05:16.710 ","End":"05:18.480","Text":"which give different answers,"},{"Start":"05:18.480 ","End":"05:20.655","Text":"and so this limit,"},{"Start":"05:20.655 ","End":"05:23.720","Text":"we conclude, does not exist."},{"Start":"05:23.720 ","End":"05:26.045","Text":"It\u0027s nonexistent."},{"Start":"05:26.045 ","End":"05:28.580","Text":"That demonstrates the technique."},{"Start":"05:28.580 ","End":"05:35.690","Text":"Don\u0027t forget that when you choose the function or the path like y equals x here,"},{"Start":"05:35.690 ","End":"05:38.870","Text":"and let\u0027s see it was y equals 2x in the second case,"},{"Start":"05:38.870 ","End":"05:44.030","Text":"that it has to contain the point in question otherwise the limits not right."},{"Start":"05:44.030 ","End":"05:47.210","Text":"For example, here, although x is going to 0,"},{"Start":"05:47.210 ","End":"05:53.315","Text":"or here also y is going to 0 because 0,0 is on the function,"},{"Start":"05:53.315 ","End":"05:55.780","Text":"then when x is 0, y is 0."},{"Start":"05:55.780 ","End":"05:58.820","Text":"You have to make sure that it contains the point."},{"Start":"05:58.820 ","End":"06:02.075","Text":"Let\u0027s take another example now."},{"Start":"06:02.075 ","End":"06:04.940","Text":"The limit as x,"},{"Start":"06:04.940 ","End":"06:06.785","Text":"y goes to 0,"},{"Start":"06:06.785 ","End":"06:16.595","Text":"0 of xy cubed over x squared plus y^6."},{"Start":"06:16.595 ","End":"06:20.015","Text":"If you try the iterated limit method,"},{"Start":"06:20.015 ","End":"06:24.080","Text":"you\u0027ll get 0 in both cases,"},{"Start":"06:24.080 ","End":"06:27.070","Text":"so we can\u0027t conclude anything."},{"Start":"06:27.070 ","End":"06:29.780","Text":"Let\u0027s try this paths method."},{"Start":"06:29.780 ","End":"06:33.485","Text":"We\u0027re going to try and find 2 paths that go to 0,0,"},{"Start":"06:33.485 ","End":"06:36.185","Text":"and we\u0027ll take the limit along each path and"},{"Start":"06:36.185 ","End":"06:39.230","Text":"find a pair of paths that give 2 different values."},{"Start":"06:39.230 ","End":"06:43.970","Text":"The first path I\u0027ll take will be y equals x."},{"Start":"06:43.970 ","End":"06:47.240","Text":"It doesn\u0027t really matter for the first one which one"},{"Start":"06:47.240 ","End":"06:50.465","Text":"you take as long as it goes through 0,0, of course."},{"Start":"06:50.465 ","End":"06:53.735","Text":"The problem might be to find the second path that\u0027s different."},{"Start":"06:53.735 ","End":"06:56.555","Text":"If we try this path,"},{"Start":"06:56.555 ","End":"07:02.450","Text":"what we do is we substitute y equals x in this limit and we get"},{"Start":"07:02.450 ","End":"07:07.760","Text":"x times x cubed over x"},{"Start":"07:07.760 ","End":"07:15.455","Text":"squared plus x^6 as x goes to 0 and then of course y goes to 0 also."},{"Start":"07:15.455 ","End":"07:21.335","Text":"This equals the limit. Well, let\u0027s see."},{"Start":"07:21.335 ","End":"07:26.745","Text":"This is x^4, and the denominator,"},{"Start":"07:26.745 ","End":"07:32.360","Text":"I can take x squared outside the brackets and"},{"Start":"07:32.360 ","End":"07:38.870","Text":"get 1 plus x^4 as x goes to 0,"},{"Start":"07:38.870 ","End":"07:45.350","Text":"x squared will cancel with x^4, giving x squared,"},{"Start":"07:45.350 ","End":"07:54.860","Text":"which equals the limit of x squared over 1 plus x squared,"},{"Start":"07:54.860 ","End":"07:58.145","Text":"x goes to 0, which equals,"},{"Start":"07:58.145 ","End":"08:05.270","Text":"I can just substitute x equals 0 and I get 0 over 1, which is 0."},{"Start":"08:05.270 ","End":"08:11.930","Text":"Another thing is to find another path that will give a different result."},{"Start":"08:11.930 ","End":"08:18.050","Text":"If we try, let\u0027s say y equals 2x,"},{"Start":"08:18.050 ","End":"08:19.250","Text":"like we did last time."},{"Start":"08:19.250 ","End":"08:21.800","Text":"This was the green path and this was the red path,"},{"Start":"08:21.800 ","End":"08:27.070","Text":"we\u0027ll get the limit."},{"Start":"08:27.500 ","End":"08:30.420","Text":"Let\u0027s see, y is 2x,"},{"Start":"08:30.420 ","End":"08:34.395","Text":"so we get x times 2x"},{"Start":"08:34.395 ","End":"08:42.850","Text":"cubed over x squared plus 2x^6."},{"Start":"08:43.040 ","End":"08:47.430","Text":"Let\u0027s see, 2 cubed is 8,"},{"Start":"08:47.430 ","End":"08:57.755","Text":"and we\u0027ll get x^4 over x squared can come outside the brackets,1 plus,"},{"Start":"08:57.755 ","End":"09:04.980","Text":"now 2x^6 is 64x^6."},{"Start":"09:04.980 ","End":"09:08.475","Text":"I only write x^4 because I\u0027m taking the x squared out."},{"Start":"09:08.475 ","End":"09:12.630","Text":"I\u0027m sorry, this is the limit,"},{"Start":"09:12.630 ","End":"09:15.985","Text":"I just forgot to write limit."},{"Start":"09:15.985 ","End":"09:19.340","Text":"But you see that if I cancel the x squared,"},{"Start":"09:19.340 ","End":"09:24.559","Text":"I\u0027ll get x squared over something plus something."},{"Start":"09:24.559 ","End":"09:26.765","Text":"Again, this will be 0."},{"Start":"09:26.765 ","End":"09:30.590","Text":"That doesn\u0027t help us because we\u0027ve got the same numbers."},{"Start":"09:30.590 ","End":"09:32.490","Text":"The question is what to try."},{"Start":"09:32.490 ","End":"09:34.760","Text":"There\u0027s many possibilities. Like I said,"},{"Start":"09:34.760 ","End":"09:37.400","Text":"we can have y equals x, y equals 2x."},{"Start":"09:37.400 ","End":"09:41.020","Text":"You could have, y equals 4x,"},{"Start":"09:41.020 ","End":"09:43.335","Text":"y equals x squared."},{"Start":"09:43.335 ","End":"09:45.180","Text":"This goes through 0,0,"},{"Start":"09:45.180 ","End":"09:48.745","Text":"y equals square root of x goes through 0,0,"},{"Start":"09:48.745 ","End":"09:52.055","Text":"y equals e to the x minus 1,"},{"Start":"09:52.055 ","End":"09:54.350","Text":"y equals sine x."},{"Start":"09:54.350 ","End":"09:56.480","Text":"The next is 0, y is 0,"},{"Start":"09:56.480 ","End":"10:00.645","Text":"y equals natural log of x plus 1,"},{"Start":"10:00.645 ","End":"10:03.170","Text":"when x is 0, y is 0, etc."},{"Start":"10:03.170 ","End":"10:04.715","Text":"There\u0027s many to choose from."},{"Start":"10:04.715 ","End":"10:08.215","Text":"I checked a few of these and most of them actually do come out 0."},{"Start":"10:08.215 ","End":"10:10.565","Text":"How do you find something that\u0027s not 0?"},{"Start":"10:10.565 ","End":"10:14.670","Text":"Well, there is a technique that can help."},{"Start":"10:14.670 ","End":"10:17.620","Text":"The main idea is this."},{"Start":"10:17.620 ","End":"10:24.940","Text":"Try to substitute y equals something that will give us all the same degree in x."},{"Start":"10:24.940 ","End":"10:26.830","Text":"Like here we got a mixture,"},{"Start":"10:26.830 ","End":"10:30.515","Text":"we got x^4, we got x squared,"},{"Start":"10:30.515 ","End":"10:32.875","Text":"and we got x^6."},{"Start":"10:32.875 ","End":"10:37.510","Text":"What I want to do is try to get everything of the same degree. If I have the same degree."},{"Start":"10:37.510 ","End":"10:39.220","Text":"This is with x^4 only,"},{"Start":"10:39.220 ","End":"10:40.930","Text":"and this was with x^4 only,"},{"Start":"10:40.930 ","End":"10:42.985","Text":"then the x to the fourth will cancel,"},{"Start":"10:42.985 ","End":"10:47.465","Text":"and then we\u0027ll maybe get something different than 0."},{"Start":"10:47.465 ","End":"10:50.095","Text":"Let\u0027s see what we could possibly substitute."},{"Start":"10:50.095 ","End":"10:54.130","Text":"Well, the x is certainly going to be to the power of 2,"},{"Start":"10:54.130 ","End":"11:00.080","Text":"so how about if I try and make y^6 to be something in terms of x squared,"},{"Start":"11:00.080 ","End":"11:04.085","Text":"I suggest trying y equals the cube root of x."},{"Start":"11:04.085 ","End":"11:11.355","Text":"Let\u0027s try y equals the cube root of x."},{"Start":"11:11.355 ","End":"11:14.999","Text":"Now, the limit I get,"},{"Start":"11:14.999 ","End":"11:21.035","Text":"x goes to 0 of velocity original."},{"Start":"11:21.035 ","End":"11:28.340","Text":"There it is, of x times the cube root of"},{"Start":"11:28.340 ","End":"11:34.860","Text":"x cubed over x"},{"Start":"11:34.860 ","End":"11:40.750","Text":"squared plus the cube root of x^6."},{"Start":"11:40.850 ","End":"11:45.960","Text":"Now, this equals the limit x cube root,"},{"Start":"11:45.960 ","End":"11:47.700","Text":"and then cubed is just x,"},{"Start":"11:47.700 ","End":"11:51.480","Text":"x with x gives me x squared."},{"Start":"11:51.480 ","End":"11:56.550","Text":"On the bottom, x squared plus the cube root of x^6"},{"Start":"11:56.550 ","End":"12:02.415","Text":"is x^6 over 3 is just x squared."},{"Start":"12:02.415 ","End":"12:06.465","Text":"Now, this over this is 1/2,"},{"Start":"12:06.465 ","End":"12:10.050","Text":"so the limit is just 1/2."},{"Start":"12:10.050 ","End":"12:13.100","Text":"It\u0027s x squared over 2 x squared and the x squared cancels,"},{"Start":"12:13.100 ","End":"12:14.570","Text":"so it gives me 1/2."},{"Start":"12:14.570 ","End":"12:22.350","Text":"Now, 1/2 is certainly different from 0."},{"Start":"12:22.350 ","End":"12:24.560","Text":"Because they are different,"},{"Start":"12:24.560 ","End":"12:28.024","Text":"I can now conclude that I have 2 different paths."},{"Start":"12:28.024 ","End":"12:31.630","Text":"The path y equals x,"},{"Start":"12:31.630 ","End":"12:34.445","Text":"and the path y equals cube root of x."},{"Start":"12:34.445 ","End":"12:37.010","Text":"There\u0027s 2 ways of going to 0,0,"},{"Start":"12:37.010 ","End":"12:39.125","Text":"and they give 2 different results,"},{"Start":"12:39.125 ","End":"12:45.540","Text":"and so this thing doesn\u0027t have a limit,"},{"Start":"12:45.540 ","End":"12:52.365","Text":"no limit or this thing is non-existent, does not exist."},{"Start":"12:52.365 ","End":"12:54.895","Text":"We\u0027re done with this example."},{"Start":"12:54.895 ","End":"13:00.620","Text":"I\u0027d like to remark that this technique with the limit along a path"},{"Start":"13:00.620 ","End":"13:02.465","Text":"is very powerful and pretty much"},{"Start":"13:02.465 ","End":"13:06.305","Text":"any non-existent limit can be solved with this technique."},{"Start":"13:06.305 ","End":"13:11.870","Text":"On the other hand, the iterative limit is very easy so really I would suggest,"},{"Start":"13:11.870 ","End":"13:15.165","Text":"start off with the iterative because it\u0027s quick."},{"Start":"13:15.165 ","End":"13:19.040","Text":"If that\u0027s good, then fine, we\u0027re done."},{"Start":"13:19.040 ","End":"13:21.560","Text":"But if it\u0027s not, then you can try to prove"},{"Start":"13:21.560 ","End":"13:25.495","Text":"nonexistence with the limit along a path method."},{"Start":"13:25.495 ","End":"13:30.815","Text":"Also, remind you that there are exercises following the tutorial."},{"Start":"13:30.815 ","End":"13:32.510","Text":"I forget which number question,"},{"Start":"13:32.510 ","End":"13:36.870","Text":"but you can find them and practice. I\u0027m done."}],"ID":8885},{"Watched":false,"Name":"Choosing the Right Technique","Duration":"9m 11s","ChapterTopicVideoID":8512,"CourseChapterTopicPlaylistID":4970,"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":"In this clip, I\u0027ll be talking about how to choose"},{"Start":"00:03.570 ","End":"00:06.840","Text":"the right technique or combination of techniques."},{"Start":"00:06.840 ","End":"00:13.815","Text":"In short, a strategy for solving problems about limit of a function of 2 variables."},{"Start":"00:13.815 ","End":"00:16.230","Text":"Up to now, we\u0027ve learned 4 techniques,"},{"Start":"00:16.230 ","End":"00:19.755","Text":"and let me just reiterate them."},{"Start":"00:19.755 ","End":"00:22.110","Text":"We had technique number 1,"},{"Start":"00:22.110 ","End":"00:27.990","Text":"which was substitution, and I\u0027ll call it regular substitution because there was another."},{"Start":"00:27.990 ","End":"00:30.885","Text":"Number 1, regular substitution,"},{"Start":"00:30.885 ","End":"00:35.500","Text":"number 2 was polar substitution,"},{"Start":"00:35.500 ","End":"00:43.085","Text":"number 3 was a technique for non-existence using iterated limits."},{"Start":"00:43.085 ","End":"00:46.070","Text":"We do the iterated limits in 2 ways,"},{"Start":"00:46.070 ","End":"00:47.900","Text":"and if we get different answers,"},{"Start":"00:47.900 ","End":"00:50.760","Text":"then we don\u0027t have a limit,"},{"Start":"00:50.760 ","End":"01:00.075","Text":"and technique number 4 was also non-existence involving"},{"Start":"01:00.075 ","End":"01:03.780","Text":"paths to find 2 paths to"},{"Start":"01:03.780 ","End":"01:09.815","Text":"the limit point and to show that we get 2 different answers along 2 different paths."},{"Start":"01:09.815 ","End":"01:13.685","Text":"I\u0027ll take an example just to help see what\u0027s going on."},{"Start":"01:13.685 ","End":"01:15.680","Text":"In almost all cases,"},{"Start":"01:15.680 ","End":"01:21.605","Text":"I would start with technique number 3, the iterated limits."},{"Start":"01:21.605 ","End":"01:25.755","Text":"The reason is it\u0027s so quick to do,"},{"Start":"01:25.755 ","End":"01:30.935","Text":"and you might just get lucky and be able to end the process right there."},{"Start":"01:30.935 ","End":"01:35.735","Text":"Let\u0027s try this 1 with the 2 different iterated limits."},{"Start":"01:35.735 ","End":"01:40.580","Text":"If I take x equals 0 first and then y equals 0, what do I get?"},{"Start":"01:40.580 ","End":"01:44.569","Text":"If x is 0, then this and this disappear."},{"Start":"01:44.569 ","End":"01:47.870","Text":"I get y squared over y, which is y,"},{"Start":"01:47.870 ","End":"01:49.400","Text":"and then when y goes to 0,"},{"Start":"01:49.400 ","End":"01:53.640","Text":"this equals 0, so that\u0027s 1 way I get 0."},{"Start":"01:53.640 ","End":"01:56.540","Text":"The other way, I let y equals 0 first,"},{"Start":"01:56.540 ","End":"01:57.860","Text":"I get x squared over x,"},{"Start":"01:57.860 ","End":"02:00.620","Text":"which is x, and then let x go to 0,"},{"Start":"02:00.620 ","End":"02:02.750","Text":"and so I get 0."},{"Start":"02:02.750 ","End":"02:05.930","Text":"So just talking on your heads, in a few seconds,"},{"Start":"02:05.930 ","End":"02:09.160","Text":"you can get the iterated limits."},{"Start":"02:09.160 ","End":"02:11.435","Text":"Now, in this case, they happen to be equal,"},{"Start":"02:11.435 ","End":"02:13.010","Text":"so it doesn\u0027t help us,"},{"Start":"02:13.010 ","End":"02:14.615","Text":"so we need to continue."},{"Start":"02:14.615 ","End":"02:16.490","Text":"But if we did get 2 different numbers,"},{"Start":"02:16.490 ","End":"02:19.850","Text":"we could stop right away and say, there is no limit."},{"Start":"02:19.850 ","End":"02:26.385","Text":"Next after this, I would try then number 1,"},{"Start":"02:26.385 ","End":"02:35.065","Text":"the regular substitution if possible."},{"Start":"02:35.065 ","End":"02:37.090","Text":"But looking at this,"},{"Start":"02:37.090 ","End":"02:43.815","Text":"that doesn\u0027t seem to be anything I could substitute like t equals x plus y,"},{"Start":"02:43.815 ","End":"02:48.160","Text":"it doesn\u0027t help because I don\u0027t know how to express this in terms of"},{"Start":"02:48.160 ","End":"02:51.705","Text":"t. I\u0027ve looked at it a"},{"Start":"02:51.705 ","End":"02:57.040","Text":"short while and it doesn\u0027t seem to be anything to substitute for t,"},{"Start":"02:57.040 ","End":"03:01.240","Text":"but I\u0027d like to give an example where regular substitution would work."},{"Start":"03:01.240 ","End":"03:08.784","Text":"Let\u0027s say we have the limit x goes to 0 and y goes to 0."},{"Start":"03:08.784 ","End":"03:18.065","Text":"The limit as x and y go both to 1 of log x plus log y over x, y minus 1."},{"Start":"03:18.065 ","End":"03:20.270","Text":"Notice that if I just substitute,"},{"Start":"03:20.270 ","End":"03:24.770","Text":"I get 0 plus 0 over 1 minus 1,"},{"Start":"03:24.770 ","End":"03:26.930","Text":"so it\u0027s a 0 over 0."},{"Start":"03:26.930 ","End":"03:29.965","Text":"If you\u0027re even a bit sharp-eyed,"},{"Start":"03:29.965 ","End":"03:33.785","Text":"you\u0027ll see right away that what we do is rewrite"},{"Start":"03:33.785 ","End":"03:39.440","Text":"the numerator as a natural log of xy using"},{"Start":"03:39.440 ","End":"03:43.705","Text":"logarithm rules over xy minus"},{"Start":"03:43.705 ","End":"03:49.640","Text":"1 and then a substitution of t equals xy will take care of it,"},{"Start":"03:49.640 ","End":"03:55.620","Text":"because then you\u0027ll get natural log of t over t minus 1 and you could handle that."},{"Start":"03:55.690 ","End":"03:59.570","Text":"Here, regular substitution worked nicely,"},{"Start":"03:59.570 ","End":"04:01.520","Text":"but here we can\u0027t seem to find anything,"},{"Start":"04:01.520 ","End":"04:03.455","Text":"so let\u0027s continue with this 1."},{"Start":"04:03.455 ","End":"04:08.240","Text":"If number 3 and number 1 don\u0027t work for us,"},{"Start":"04:08.240 ","End":"04:10.550","Text":"the next thing to try, I believe,"},{"Start":"04:10.550 ","End":"04:15.195","Text":"is number 2, the polar substitution."},{"Start":"04:15.195 ","End":"04:16.520","Text":"Just to remind you,"},{"Start":"04:16.520 ","End":"04:25.785","Text":"we substitute x equals r cosine Theta and y equals r sine Theta,"},{"Start":"04:25.785 ","End":"04:34.095","Text":"and we replace the going to 0 with r goes to 0 from the right,"},{"Start":"04:34.095 ","End":"04:39.560","Text":"and we also remember that x squared plus y squared is r squared."},{"Start":"04:39.560 ","End":"04:48.380","Text":"We get the limit as r goes to 0 from the right"},{"Start":"04:48.380 ","End":"04:57.945","Text":"of x squared plus y squared is r squared over r cosine Theta plus r sine Theta."},{"Start":"04:57.945 ","End":"05:00.260","Text":"I can take r outside the bracket,"},{"Start":"05:00.260 ","End":"05:02.465","Text":"so it will cancel with 1 of the r\u0027s,"},{"Start":"05:02.465 ","End":"05:07.785","Text":"so I get the limit as r goes to 0 of"},{"Start":"05:07.785 ","End":"05:14.050","Text":"r over sine Theta plus cosine Theta."},{"Start":"05:14.050 ","End":"05:16.625","Text":"Now, in our previous examples,"},{"Start":"05:16.625 ","End":"05:21.605","Text":"this turned out to be something that goes to 0 times something bounded."},{"Start":"05:21.605 ","End":"05:25.340","Text":"But that\u0027s if the sine Theta plus cosine Theta was in the numerator,"},{"Start":"05:25.340 ","End":"05:27.139","Text":"then I could say it\u0027s bounded."},{"Start":"05:27.139 ","End":"05:29.735","Text":"But if it\u0027s in the denominator,"},{"Start":"05:29.735 ","End":"05:33.955","Text":"it\u0027s actually not bounded because the denominator could be"},{"Start":"05:33.955 ","End":"05:39.545","Text":"0 if you let Theta equal 135 degrees,"},{"Start":"05:39.545 ","End":"05:41.430","Text":"whatever that is in radians,"},{"Start":"05:41.430 ","End":"05:43.605","Text":"what is it, 3 Pi over 4."},{"Start":"05:43.605 ","End":"05:48.980","Text":"At 135 degrees, the sine is equal to 1"},{"Start":"05:48.980 ","End":"05:54.740","Text":"over root 2 and the cosine is minus 1 over root 2 so this could get to 0."},{"Start":"05:54.740 ","End":"05:59.210","Text":"The expression is undefined sometimes,"},{"Start":"05:59.210 ","End":"06:02.690","Text":"so this doesn\u0027t really get me anywhere."},{"Start":"06:02.690 ","End":"06:07.730","Text":"The final thing to do is to go for technique number 4,"},{"Start":"06:07.730 ","End":"06:12.725","Text":"which is actually the most powerful technique if not the easiest."},{"Start":"06:12.725 ","End":"06:15.940","Text":"Let\u0027s see how we would do this with paths."},{"Start":"06:15.940 ","End":"06:19.515","Text":"But first, there\u0027s something I forgot to say in polar substitution,"},{"Start":"06:19.515 ","End":"06:25.965","Text":"and that is that it only works when the limit is at 0, 0."},{"Start":"06:25.965 ","End":"06:29.435","Text":"Otherwise, you can\u0027t use the polar substitution. Back here."},{"Start":"06:29.435 ","End":"06:31.325","Text":"To do it with paths,"},{"Start":"06:31.325 ","End":"06:35.660","Text":"I almost always choose y equals x to be"},{"Start":"06:35.660 ","End":"06:42.260","Text":"my first path that I try because it\u0027s very easy to substitute,"},{"Start":"06:42.260 ","End":"06:45.110","Text":"and also, it doesn\u0027t matter what you choose first."},{"Start":"06:45.110 ","End":"06:49.225","Text":"The difficulty is usually finding another 1 that\u0027s going to be different."},{"Start":"06:49.225 ","End":"06:51.350","Text":"If I let y equals x,"},{"Start":"06:51.350 ","End":"06:59.790","Text":"what I\u0027ll get is the limit of y is x,"},{"Start":"06:59.790 ","End":"07:02.220","Text":"so this will be 2x squared."},{"Start":"07:02.220 ","End":"07:10.785","Text":"Y is x, so this will be 2x and the 2x canceled,"},{"Start":"07:10.785 ","End":"07:19.165","Text":"so all I get is the limit of x when xy both go to 0,"},{"Start":"07:19.165 ","End":"07:23.440","Text":"and this is equal to 0."},{"Start":"07:23.600 ","End":"07:28.180","Text":"Now I have to find a path that doesn\u0027t give me 0."},{"Start":"07:28.180 ","End":"07:30.025","Text":"Just looking at it a bit,"},{"Start":"07:30.025 ","End":"07:31.775","Text":"I get the idea,"},{"Start":"07:31.775 ","End":"07:34.690","Text":"hey, what if I let y equals minus x?"},{"Start":"07:34.690 ","End":"07:37.645","Text":"I\u0027ve got a 0 on the denominator. Let\u0027s try that."},{"Start":"07:37.645 ","End":"07:40.330","Text":"Y equals minus x,"},{"Start":"07:40.330 ","End":"07:42.920","Text":"I get the limit."},{"Start":"07:44.630 ","End":"07:47.355","Text":"This comes out the same,"},{"Start":"07:47.355 ","End":"07:53.219","Text":"2x squared, but here I get over 0."},{"Start":"07:53.219 ","End":"07:55.010","Text":"Now, this is a real 0,"},{"Start":"07:55.010 ","End":"07:56.600","Text":"not just tends to 0,"},{"Start":"07:56.600 ","End":"07:58.130","Text":"this is actually over 0,"},{"Start":"07:58.130 ","End":"08:00.410","Text":"so this is undefined,"},{"Start":"08:00.410 ","End":"08:04.405","Text":"and undefined is different from 0."},{"Start":"08:04.405 ","End":"08:08.130","Text":"It doesn\u0027t have to just be 2 numbers that are different,"},{"Start":"08:08.130 ","End":"08:10.860","Text":"it could be that one\u0027s 0 and one\u0027s undefined,"},{"Start":"08:10.860 ","End":"08:13.700","Text":"or this is 0 and this goes to infinity."},{"Start":"08:13.700 ","End":"08:17.750","Text":"In any case, we get 2 different things along 2 different paths."},{"Start":"08:17.750 ","End":"08:23.210","Text":"Finally, we conclude that this has no limit."},{"Start":"08:23.210 ","End":"08:26.150","Text":"The limit doesn\u0027t exist."},{"Start":"08:26.150 ","End":"08:29.340","Text":"That\u0027s about it."},{"Start":"08:29.500 ","End":"08:31.550","Text":"I would like to add though,"},{"Start":"08:31.550 ","End":"08:34.805","Text":"that sometimes life can be made easier."},{"Start":"08:34.805 ","End":"08:39.170","Text":"Say this was an exam question and the question said,"},{"Start":"08:39.170 ","End":"08:45.560","Text":"\"Prove that the limit of this function at such and such a point is equal to 4\","},{"Start":"08:45.560 ","End":"08:47.974","Text":"then it tells you that the limit exists,"},{"Start":"08:47.974 ","End":"08:50.660","Text":"so I wouldn\u0027t bother with 3 and 4 because they\u0027re"},{"Start":"08:50.660 ","End":"08:53.710","Text":"just for non-existence I just focus on 1 and 2."},{"Start":"08:53.710 ","End":"08:55.420","Text":"On the other hand, if the question was,"},{"Start":"08:55.420 ","End":"08:58.330","Text":"\"Prove that this limit does not exist\","},{"Start":"08:58.330 ","End":"09:00.865","Text":"then I would just use 3 and 4."},{"Start":"09:00.865 ","End":"09:03.785","Text":"Sometimes, you can shorten it a bit."},{"Start":"09:03.785 ","End":"09:08.899","Text":"That\u0027s general wisdom of what strategy,"},{"Start":"09:08.899 ","End":"09:10.220","Text":"what techniques to use,"},{"Start":"09:10.220 ","End":"09:12.300","Text":"and I\u0027m done here."}],"ID":8886},{"Watched":false,"Name":"Continuity of a Function of Two Variables","Duration":"5m 1s","ChapterTopicVideoID":8513,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.625","Text":"In this clip, I\u0027ll be talking about continuity of a function of 2 variables."},{"Start":"00:05.625 ","End":"00:09.795","Text":"Let\u0027s first remember what it means for 1 variable."},{"Start":"00:09.795 ","End":"00:13.915","Text":"If we have a function f of x,"},{"Start":"00:13.915 ","End":"00:19.980","Text":"then we say that it\u0027s continuous at a point,"},{"Start":"00:19.980 ","End":"00:26.945","Text":"say x equals a, if the limit"},{"Start":"00:26.945 ","End":"00:35.710","Text":"as x goes to a of f of x is equal to f of a."},{"Start":"00:35.710 ","End":"00:40.370","Text":"This definition relies heavily on concept of limit."},{"Start":"00:40.370 ","End":"00:43.910","Text":"Now, that we know how to do limits in 2-dimensions,"},{"Start":"00:43.910 ","End":"00:51.395","Text":"we can take the analogy of this in 2-dimensions and say that f of x,"},{"Start":"00:51.395 ","End":"00:57.170","Text":"y is continuous at a point a,"},{"Start":"00:57.170 ","End":"01:03.110","Text":"b if the limit as x y goes to a,"},{"Start":"01:03.110 ","End":"01:06.065","Text":"b of f of x,"},{"Start":"01:06.065 ","End":"01:11.100","Text":"y is f of a, b."},{"Start":"01:11.100 ","End":"01:15.830","Text":"So here we have a definition, continuous at a point, if the limit,"},{"Start":"01:15.830 ","End":"01:19.925","Text":"as we approach that point, is equal to the value at the point."},{"Start":"01:19.925 ","End":"01:24.845","Text":"Let\u0027s take an example f of x,"},{"Start":"01:24.845 ","End":"01:29.850","Text":"y is equal to,"},{"Start":"01:29.920 ","End":"01:36.470","Text":"piecewise-defined, it\u0027ll be equal to the sine of x"},{"Start":"01:36.470 ","End":"01:43.415","Text":"squared plus y squared over x squared plus y squared,"},{"Start":"01:43.415 ","End":"01:45.800","Text":"and this is if x,"},{"Start":"01:45.800 ","End":"01:48.720","Text":"y is not the origin,"},{"Start":"01:49.010 ","End":"01:53.860","Text":"and it\u0027s going to equal 4 at the origin."},{"Start":"01:53.860 ","End":"01:58.965","Text":"If x, y equals 0, 0,"},{"Start":"01:58.965 ","End":"02:01.305","Text":"and the question is,"},{"Start":"02:01.305 ","End":"02:06.510","Text":"is it continuous at 0, 0?"},{"Start":"02:06.510 ","End":"02:09.130","Text":"Let\u0027s see,"},{"Start":"02:09.340 ","End":"02:12.110","Text":"let\u0027s take first of all,"},{"Start":"02:12.110 ","End":"02:14.645","Text":"the limit, this part."},{"Start":"02:14.645 ","End":"02:23.490","Text":"So the limit as x goes to 0 and y goes to 0 of f of x, y."},{"Start":"02:23.780 ","End":"02:27.300","Text":"Now, if x and y are going to 0, 0,"},{"Start":"02:27.300 ","End":"02:31.115","Text":"then they\u0027re not 0, 0 which means we take our definition from here,"},{"Start":"02:31.115 ","End":"02:35.630","Text":"which is equal to the limit of sine of x"},{"Start":"02:35.630 ","End":"02:43.390","Text":"squared plus y squared over x squared plus y squared."},{"Start":"02:43.820 ","End":"02:48.470","Text":"This looks like it should be done by substitution."},{"Start":"02:48.470 ","End":"02:55.220","Text":"It\u0027s so obvious that we let t equals x squared plus y squared,"},{"Start":"02:55.220 ","End":"03:00.530","Text":"and so we get the limit as, now,"},{"Start":"03:00.530 ","End":"03:05.840","Text":"what is t go to? x squared plus y squared is 0 squared plus 0 squared is 0,"},{"Start":"03:05.840 ","End":"03:11.135","Text":"so limit as t goes to 0 of"},{"Start":"03:11.135 ","End":"03:18.995","Text":"sine t over t. Actually,"},{"Start":"03:18.995 ","End":"03:26.030","Text":"technically it should be 0 plus because x squared plus y squared is always non-negative,"},{"Start":"03:26.030 ","End":"03:27.635","Text":"it doesn\u0027t matter, either way,"},{"Start":"03:27.635 ","End":"03:30.560","Text":"this limit is equal to 1."},{"Start":"03:30.560 ","End":"03:32.675","Text":"So we have a limit,"},{"Start":"03:32.675 ","End":"03:34.400","Text":"and that limit is 1."},{"Start":"03:34.400 ","End":"03:36.470","Text":"So that\u0027s this part,"},{"Start":"03:36.470 ","End":"03:38.750","Text":"but the other side of the equation,"},{"Start":"03:38.750 ","End":"03:41.260","Text":"f of a, b, f of 0,"},{"Start":"03:41.260 ","End":"03:42.590","Text":"0 is equal to, well,"},{"Start":"03:42.590 ","End":"03:47.660","Text":"it\u0027s given to us here is equal to 4, and 1 is not equal to 4,"},{"Start":"03:47.660 ","End":"03:50.165","Text":"so this equality does not hold,"},{"Start":"03:50.165 ","End":"03:52.340","Text":"so the function is not continuous."},{"Start":"03:52.340 ","End":"03:56.090","Text":"Now, remember that if this in the 1-dimensional case,"},{"Start":"03:56.090 ","End":"03:59.100","Text":"if the limit exists, but is not equal"},{"Start":"03:59.100 ","End":"04:02.429","Text":"the value at the point, then it\u0027s called a removable discontinuity."},{"Start":"04:02.429 ","End":"04:05.235","Text":"The same in the function of 2 variables,"},{"Start":"04:05.235 ","End":"04:07.240","Text":"if this limit exists,"},{"Start":"04:07.240 ","End":"04:09.140","Text":"but just happens not to equal this value,"},{"Start":"04:09.140 ","End":"04:10.684","Text":"it\u0027s also called removable."},{"Start":"04:10.684 ","End":"04:14.090","Text":"The reason it\u0027s called removable is that, if we change"},{"Start":"04:14.090 ","End":"04:17.535","Text":"the definition, and instead of the value 4 here,"},{"Start":"04:17.535 ","End":"04:19.125","Text":"we put the value 1,"},{"Start":"04:19.125 ","End":"04:21.705","Text":"then we would get a continuous function."},{"Start":"04:21.705 ","End":"04:26.960","Text":"A variation on this question, that could appear in the exam, would be to say,"},{"Start":"04:26.960 ","End":"04:29.390","Text":"given the function f as follows,"},{"Start":"04:29.390 ","End":"04:36.200","Text":"but here they give us a parameter value k, they"},{"Start":"04:36.200 ","End":"04:43.325","Text":"ask, what would be the value of k in order that this function be continuous at 0, 0."},{"Start":"04:43.325 ","End":"04:50.645","Text":"Then you would compute this and see that it comes out to be 1,"},{"Start":"04:50.645 ","End":"04:52.535","Text":"and then you would say, well,"},{"Start":"04:52.535 ","End":"04:54.155","Text":"if k equals 1,"},{"Start":"04:54.155 ","End":"04:58.820","Text":"then the function is continuous, because then that side is also 1."},{"Start":"04:58.820 ","End":"05:01.890","Text":"That\u0027s all for this clip."}],"ID":8887},{"Watched":false,"Name":"Exercise 1 part a","Duration":"3m 39s","ChapterTopicVideoID":8514,"CourseChapterTopicPlaylistID":4970,"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\u0027re given the following function of two variables,"},{"Start":"00:03.930 ","End":"00:06.285","Text":"x and y to y over x."},{"Start":"00:06.285 ","End":"00:08.039","Text":"We have to find the domain,"},{"Start":"00:08.039 ","End":"00:09.389","Text":"to sketch the domain,"},{"Start":"00:09.389 ","End":"00:11.190","Text":"and then the level curves."},{"Start":"00:11.190 ","End":"00:12.600","Text":"Start off with the domain."},{"Start":"00:12.600 ","End":"00:14.565","Text":"First of all, what is the domain?"},{"Start":"00:14.565 ","End":"00:16.455","Text":"What can go wrong here?"},{"Start":"00:16.455 ","End":"00:20.685","Text":"Well, the only thing that\u0027s a problem is a 0 in the denominator."},{"Start":"00:20.685 ","End":"00:28.960","Text":"Immediately we see that the domain could be described as x not equal to 0."},{"Start":"00:29.180 ","End":"00:32.370","Text":"If we want to sketch it,"},{"Start":"00:32.370 ","End":"00:35.850","Text":"let\u0027s start out with a pair of coordinate axes,"},{"Start":"00:35.850 ","End":"00:38.095","Text":"x and y, and let\u0027s see."},{"Start":"00:38.095 ","End":"00:44.419","Text":"It actually might be easier to say which x are not allowed."},{"Start":"00:44.419 ","End":"00:48.685","Text":"Other words, where is x equal to 0, the bad axis?"},{"Start":"00:48.685 ","End":"00:51.720","Text":"x equals 0 is precisely the y-axis,"},{"Start":"00:51.720 ","End":"00:56.085","Text":"so these are the bad values."},{"Start":"00:56.085 ","End":"00:58.640","Text":"The domain is everything that\u0027s left."},{"Start":"00:58.640 ","End":"01:01.715","Text":"It\u0027s all this half of the plane,"},{"Start":"01:01.715 ","End":"01:05.120","Text":"and this half of the plane executes the sloppiness,"},{"Start":"01:05.120 ","End":"01:07.390","Text":"you do get the idea."},{"Start":"01:07.390 ","End":"01:15.755","Text":"We can describe it as the plane without the y-axis."},{"Start":"01:15.755 ","End":"01:21.365","Text":"We\u0027re excluding the y axis."},{"Start":"01:21.365 ","End":"01:24.560","Text":"That would be in words."},{"Start":"01:24.560 ","End":"01:28.175","Text":"I\u0027m not happy with these squiggly lines I\u0027ll re-sketch it."},{"Start":"01:28.175 ","End":"01:32.855","Text":"Anyway, plane without the y-axis. That\u0027s the domain."},{"Start":"01:32.855 ","End":"01:37.410","Text":"Now what about the level curves?"},{"Start":"01:37.550 ","End":"01:42.110","Text":"For level curves, we take the value of f and make it a constant."},{"Start":"01:42.110 ","End":"01:48.110","Text":"In other words, we would get something like y over x equals k,"},{"Start":"01:48.110 ","End":"01:52.280","Text":"where different values of k give us different level curves."},{"Start":"01:52.280 ","End":"01:53.870","Text":"If you think about this,"},{"Start":"01:53.870 ","End":"01:56.525","Text":"this is just the equation,"},{"Start":"01:56.525 ","End":"02:02.640","Text":"y equals kx, which is a line through the origin."},{"Start":"02:02.710 ","End":"02:05.225","Text":"Let\u0027s just draw some of these."},{"Start":"02:05.225 ","End":"02:09.790","Text":"For example, if we take k equals 1,"},{"Start":"02:09.790 ","End":"02:13.835","Text":"then we\u0027ll get the line y equals x,"},{"Start":"02:13.835 ","End":"02:19.595","Text":"which is a 45 degree line through the origin."},{"Start":"02:19.595 ","End":"02:24.770","Text":"If we take k equals 0,"},{"Start":"02:24.770 ","End":"02:28.040","Text":"we\u0027ll get y equals 0,"},{"Start":"02:28.040 ","End":"02:32.795","Text":"which is the x-axis."},{"Start":"02:32.795 ","End":"02:34.880","Text":"If we take, say,"},{"Start":"02:34.880 ","End":"02:40.030","Text":"y equals minus 2x,"},{"Start":"02:40.030 ","End":"02:42.605","Text":"I meant k equals minus 2,"},{"Start":"02:42.605 ","End":"02:49.955","Text":"then we would get something with slope negative 2, something like this."},{"Start":"02:49.955 ","End":"02:52.670","Text":"You can get as many of these as we like."},{"Start":"02:52.670 ","End":"02:54.200","Text":"If k equals plus 2,"},{"Start":"02:54.200 ","End":"02:57.800","Text":"we get a line of slope 2 also through the origin."},{"Start":"02:57.800 ","End":"03:00.470","Text":"If k would be say, minus 1/2,"},{"Start":"03:00.470 ","End":"03:03.935","Text":"we get a slope minus 1/2 through the origin."},{"Start":"03:03.935 ","End":"03:05.900","Text":"I know they don\u0027t look really like straight lines,"},{"Start":"03:05.900 ","End":"03:07.820","Text":"but you get the idea."},{"Start":"03:07.820 ","End":"03:13.760","Text":"Note that no value of k will give us the vertical line and the y-axis is excluded,"},{"Start":"03:13.760 ","End":"03:17.840","Text":"but the other thing is, because the y-axis is excluded,"},{"Start":"03:17.840 ","End":"03:19.790","Text":"the origin has to be missing,"},{"Start":"03:19.790 ","End":"03:23.645","Text":"so actually these lines have a hole in them at the origin,"},{"Start":"03:23.645 ","End":"03:29.840","Text":"and so this is a probably more accurate diagram lines with a missing point."},{"Start":"03:29.840 ","End":"03:32.990","Text":"Other words, these are the lines y equals kx,"},{"Start":"03:32.990 ","End":"03:36.125","Text":"but x is not 0, so point missing."},{"Start":"03:36.125 ","End":"03:39.180","Text":"This is good enough and we\u0027re done."}],"ID":8888},{"Watched":false,"Name":"Exercise 1 part b","Duration":"4m 17s","ChapterTopicVideoID":8515,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:03.420","Text":"We have here a function of 2 variables,"},{"Start":"00:03.420 ","End":"00:07.260","Text":"f of x, y equals natural log of x plus natural log of y."},{"Start":"00:07.260 ","End":"00:11.340","Text":"We have to find the domain of definition and sketch it first of all."},{"Start":"00:11.340 ","End":"00:13.245","Text":"Then we\u0027ll do the level curves."},{"Start":"00:13.245 ","End":"00:16.585","Text":"First of all, we\u0027ll do the domain of definition."},{"Start":"00:16.585 ","End":"00:23.300","Text":"For the domain, we have to see which x and y or combination is not legal."},{"Start":"00:23.300 ","End":"00:25.820","Text":"Well, the only thing we have to worry about is that"},{"Start":"00:25.820 ","End":"00:28.475","Text":"the argument of the natural log has to be positive."},{"Start":"00:28.475 ","End":"00:32.880","Text":"In other words, we have to have both x positive for this to make sense"},{"Start":"00:32.880 ","End":"00:36.740","Text":"and we have to have y positive for this to make sense."},{"Start":"00:36.740 ","End":"00:38.360","Text":"Other than that, there\u0027s no restriction."},{"Start":"00:38.360 ","End":"00:40.190","Text":"Addition has no problem."},{"Start":"00:40.190 ","End":"00:44.390","Text":"If we think about this, this is just the first quadrant."},{"Start":"00:44.390 ","End":"00:49.490","Text":"But strictly positive here means that the axis are not included"},{"Start":"00:49.490 ","End":"00:53.565","Text":"and so we get something like this."},{"Start":"00:53.565 ","End":"00:57.615","Text":"But you got to be careful not to reach the axis."},{"Start":"00:57.615 ","End":"01:00.555","Text":"There we go."},{"Start":"01:00.555 ","End":"01:02.735","Text":"Anyway, get the idea."},{"Start":"01:02.735 ","End":"01:07.445","Text":"It\u0027s the positive first quadrant."},{"Start":"01:07.445 ","End":"01:12.830","Text":"Now what about the level curves?"},{"Start":"01:12.830 ","End":"01:16.880","Text":"To get these, we let f of x, y be some constant k."},{"Start":"01:16.880 ","End":"01:22.280","Text":"In other words, natural log of x plus natural log of y equals k."},{"Start":"01:22.280 ","End":"01:25.385","Text":"Now, how do we sketch these?"},{"Start":"01:25.385 ","End":"01:27.095","Text":"Let\u0027s just rewrite it a bit."},{"Start":"01:27.095 ","End":"01:29.449","Text":"Remember the property of logarithms."},{"Start":"01:29.449 ","End":"01:35.030","Text":"The sum of the logarithms is the logarithm of the product."},{"Start":"01:35.030 ","End":"01:40.640","Text":"In other words, we get that the natural log of x times y is equal to k."},{"Start":"01:40.640 ","End":"01:42.310","Text":"Now I\u0027d like you to remember"},{"Start":"01:42.310 ","End":"01:46.270","Text":"the definition of logarithm and its relationship to the exponential function."},{"Start":"01:46.270 ","End":"01:55.575","Text":"In general, if we have log to the base a of b is equal to c,"},{"Start":"01:55.575 ","End":"01:59.290","Text":"this is completely equivalent to a,"},{"Start":"01:59.290 ","End":"02:04.390","Text":"to the power of c, is equal to b."},{"Start":"02:04.550 ","End":"02:08.050","Text":"In this case, what we get is that,"},{"Start":"02:08.050 ","End":"02:11.140","Text":"well, natural log is log to the base e."},{"Start":"02:11.140 ","End":"02:23.120","Text":"So we get that e to the power of the c here is the k is equal to x, y."},{"Start":"02:23.270 ","End":"02:27.710","Text":"If I rewrite this as y in terms of x,"},{"Start":"02:27.710 ","End":"02:30.200","Text":"then switch the sides and divide by x,"},{"Start":"02:30.200 ","End":"02:37.490","Text":"I get y equals e to the k over x."},{"Start":"02:37.490 ","End":"02:40.760","Text":"Now, k is a constant and so is e to the k."},{"Start":"02:40.760 ","End":"02:44.090","Text":"In fact, when k runs through all possible constants,"},{"Start":"02:44.090 ","End":"02:49.280","Text":"e to the k actually only gets to be a positive constant"},{"Start":"02:49.280 ","End":"02:53.135","Text":"so I can actually write this as y equals,"},{"Start":"02:53.135 ","End":"02:57.505","Text":"let\u0027s say, b over x."},{"Start":"02:57.505 ","End":"03:01.550","Text":"But you have to remember that b is positive."},{"Start":"03:01.550 ","End":"03:03.950","Text":"Instead of letting k vary,"},{"Start":"03:03.950 ","End":"03:06.420","Text":"we just let b vary."},{"Start":"03:07.040 ","End":"03:10.930","Text":"Suppose I let b equals 1,"},{"Start":"03:10.930 ","End":"03:14.290","Text":"then I\u0027ll get y equals 1 over x,"},{"Start":"03:14.290 ","End":"03:15.470","Text":"and I\u0027ll sketch it in a moment."},{"Start":"03:15.470 ","End":"03:17.545","Text":"If b equals 2,"},{"Start":"03:17.545 ","End":"03:21.010","Text":"we\u0027ll get y equals 2 over x."},{"Start":"03:21.010 ","End":"03:22.960","Text":"B could also be less than 1."},{"Start":"03:22.960 ","End":"03:24.430","Text":"B could be a 1/2."},{"Start":"03:24.430 ","End":"03:30.810","Text":"Then we get y equals 1/2 over x."},{"Start":"03:30.810 ","End":"03:36.610","Text":"Now, 1 over x, we know, say this is the point 1, 1,"},{"Start":"03:36.610 ","End":"03:38.530","Text":"and it goes something like this"},{"Start":"03:38.530 ","End":"03:42.835","Text":"and it asymptotically reaches the x-axis and the y-axis."},{"Start":"03:42.835 ","End":"03:45.460","Text":"If we have 2 over x, it\u0027s bigger."},{"Start":"03:45.460 ","End":"03:48.145","Text":"It goes through the points 1, 2 and 2, 1."},{"Start":"03:48.145 ","End":"03:51.695","Text":"It will be somewhere further out here."},{"Start":"03:51.695 ","End":"03:56.485","Text":"If we have something b is less than 1,"},{"Start":"03:56.485 ","End":"03:58.310","Text":"say a 1/2 over x,"},{"Start":"03:58.310 ","End":"04:01.865","Text":"then it will go through a point a half-1 and 1 a 1/2."},{"Start":"04:01.865 ","End":"04:04.620","Text":"It will be something like this."},{"Start":"04:05.380 ","End":"04:10.280","Text":"Also, each of them asymptotically reaches the y and the x-axis."},{"Start":"04:10.280 ","End":"04:15.170","Text":"I think this will do to give the idea of the level curves."},{"Start":"04:15.170 ","End":"04:17.790","Text":"We\u0027re done."}],"ID":8889},{"Watched":false,"Name":"Exercise 1 part c","Duration":"2m 42s","ChapterTopicVideoID":8516,"CourseChapterTopicPlaylistID":4970,"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":"Here, we\u0027re given the function of 2 variables, x and y,"},{"Start":"00:03.900 ","End":"00:08.115","Text":"x squared plus y squared to find its domain first,"},{"Start":"00:08.115 ","End":"00:09.855","Text":"and to sketch the domain,"},{"Start":"00:09.855 ","End":"00:12.165","Text":"and then the level curves."},{"Start":"00:12.165 ","End":"00:14.700","Text":"Let\u0027s start with the domain."},{"Start":"00:14.700 ","End":"00:18.330","Text":"There\u0027s actually no restriction on x and y."},{"Start":"00:18.330 ","End":"00:20.190","Text":"Any number can be squared,"},{"Start":"00:20.190 ","End":"00:24.210","Text":"x or y, and we can add numbers. There\u0027s no problem."},{"Start":"00:24.210 ","End":"00:29.475","Text":"It\u0027s really all x, y,"},{"Start":"00:29.475 ","End":"00:33.670","Text":"or if we want to interpret that it would be the whole plane,"},{"Start":"00:33.950 ","End":"00:39.250","Text":"and it\u0027s a bit silly to sketch it, but we can do it."},{"Start":"00:39.250 ","End":"00:44.850","Text":"We just sketch everything, the whole plane."},{"Start":"00:44.990 ","End":"00:50.125","Text":"That\u0027s domain. Now, how about level curves?"},{"Start":"00:50.125 ","End":"00:54.860","Text":"For level curves, we let the value of the function be a constant k,"},{"Start":"00:54.860 ","End":"01:00.140","Text":"so we would get x squared plus y squared equals k. Now,"},{"Start":"01:00.140 ","End":"01:03.935","Text":"notice that this is non-negative and this is non-negative,"},{"Start":"01:03.935 ","End":"01:07.505","Text":"so our k has to be bigger or equal to 0."},{"Start":"01:07.505 ","End":"01:09.170","Text":"No point taking a negative k,"},{"Start":"01:09.170 ","End":"01:10.730","Text":"we just won\u0027t get a level curve."},{"Start":"01:10.730 ","End":"01:14.480","Text":"This reminds me of a circle."},{"Start":"01:14.480 ","End":"01:16.910","Text":"In fact, if k is bigger or equal to 0,"},{"Start":"01:16.910 ","End":"01:20.750","Text":"it has a non-negative square root r,"},{"Start":"01:20.750 ","End":"01:28.115","Text":"so let\u0027s just write this as x squared plus y squared equal r squared,"},{"Start":"01:28.115 ","End":"01:31.760","Text":"where r is also bigger or equal to 0."},{"Start":"01:31.760 ","End":"01:34.925","Text":"It doesn\u0027t have to be, but I want it to be bigger or equal to 0,"},{"Start":"01:34.925 ","End":"01:38.840","Text":"and that will give us all the possible ks and that will be the radii."},{"Start":"01:38.840 ","End":"01:42.600","Text":"For example, if I let r equal 1,"},{"Start":"01:42.820 ","End":"01:49.340","Text":"then I\u0027ll get x squared plus y squared equals 1 or 1 squared."},{"Start":"01:49.340 ","End":"01:51.800","Text":"If I let r equal 2,"},{"Start":"01:51.800 ","End":"01:56.660","Text":"I\u0027ll get x squared plus y squared equals 4 or 2 squared."},{"Start":"01:56.660 ","End":"01:59.105","Text":"I can also let r equals 0."},{"Start":"01:59.105 ","End":"02:05.675","Text":"If r equals 0, I get x squared plus y squared equals 0 squared, which is 0."},{"Start":"02:05.675 ","End":"02:07.790","Text":"Actually, this is the most interesting 1,"},{"Start":"02:07.790 ","End":"02:09.940","Text":"but I\u0027ll start with this 1."},{"Start":"02:09.940 ","End":"02:13.490","Text":"Here we have a circle of radius 1,"},{"Start":"02:13.490 ","End":"02:15.620","Text":"might look something like this."},{"Start":"02:15.620 ","End":"02:18.830","Text":"Here, we\u0027d have a circle of radius 2,"},{"Start":"02:18.830 ","End":"02:21.800","Text":"might look something like this."},{"Start":"02:21.800 ","End":"02:25.220","Text":"Here, circle of radius 0, in fact,"},{"Start":"02:25.220 ","End":"02:28.840","Text":"the only possible x and y that will make this 0 is x and y are both 0,"},{"Start":"02:28.840 ","End":"02:34.750","Text":"so this point itself is actually a level curve for 0 and so on,"},{"Start":"02:34.750 ","End":"02:39.635","Text":"all the possible circles are all the level curves."},{"Start":"02:39.635 ","End":"02:43.110","Text":"That\u0027s it. We\u0027re done."}],"ID":8890},{"Watched":false,"Name":"Exercise 1 part d","Duration":"3m 29s","ChapterTopicVideoID":8517,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:04.650","Text":"Here we\u0027re given a function of 2 variables. This is it."},{"Start":"00:04.650 ","End":"00:08.100","Text":"We want the domain of definition including a sketch,"},{"Start":"00:08.100 ","End":"00:11.220","Text":"and we also want the sketch of the level curves."},{"Start":"00:11.220 ","End":"00:14.805","Text":"Let\u0027s look first of all at the domain."},{"Start":"00:14.805 ","End":"00:16.785","Text":"Now what can go wrong?"},{"Start":"00:16.785 ","End":"00:20.505","Text":"The only thing is that there\u0027s a square root here and under the square root,"},{"Start":"00:20.505 ","End":"00:22.380","Text":"we have to have something non-negative."},{"Start":"00:22.380 ","End":"00:26.655","Text":"In other words, we get that 1 minus x squared minus y squared."},{"Start":"00:26.655 ","End":"00:29.595","Text":"Non-negative means bigger or equal to 0."},{"Start":"00:29.595 ","End":"00:31.515","Text":"If you just play around with this,"},{"Start":"00:31.515 ","End":"00:37.960","Text":"you\u0027ll see that if we get that x squared plus y squared less than or equal to 1."},{"Start":"00:38.450 ","End":"00:44.595","Text":"If x squared plus y squared equals 1, that\u0027s the circle."},{"Start":"00:44.595 ","End":"00:47.955","Text":"This might be x squared plus y squared equals 1."},{"Start":"00:47.955 ","End":"00:50.930","Text":"Less than or equal to 1 means that it\u0027s on"},{"Start":"00:50.930 ","End":"00:54.950","Text":"a circle of smaller radius so it\u0027s the inside of"},{"Start":"00:54.950 ","End":"00:58.790","Text":"the circle including the boundary of the circle so it\u0027s"},{"Start":"00:58.790 ","End":"01:03.395","Text":"the whole disk including the boundary."},{"Start":"01:03.395 ","End":"01:05.179","Text":"That\u0027s the domain."},{"Start":"01:05.179 ","End":"01:08.869","Text":"Now, what about the level curves?"},{"Start":"01:08.869 ","End":"01:12.950","Text":"In this case, we let f of x and y equal some constant k"},{"Start":"01:12.950 ","End":"01:16.470","Text":"so we get the square root of 1 minus x"},{"Start":"01:16.470 ","End":"01:24.874","Text":"squared minus y squared equals k. If you square both sides and change sides,"},{"Start":"01:24.874 ","End":"01:32.665","Text":"we get that x squared plus y squared is equal to 1 minus k squared."},{"Start":"01:32.665 ","End":"01:37.490","Text":"Now, k being a square root has to be bigger or equal to 0."},{"Start":"01:37.490 ","End":"01:40.610","Text":"Because of the restriction that"},{"Start":"01:40.610 ","End":"01:44.230","Text":"x squared plus y squared has to be less than or equal to 1,"},{"Start":"01:44.230 ","End":"01:49.010","Text":"k also has to be less than or equal to 1,"},{"Start":"01:49.010 ","End":"01:51.500","Text":"because if k is any bigger than 1,"},{"Start":"01:51.500 ","End":"01:53.330","Text":"then this becomes negative."},{"Start":"01:53.330 ","End":"01:55.925","Text":"Actually I could say that,"},{"Start":"01:55.925 ","End":"02:03.995","Text":"I\u0027ll erase this and I\u0027ll write it as k is between 0 and 1."},{"Start":"02:03.995 ","End":"02:05.980","Text":"Let\u0027s try different values."},{"Start":"02:05.980 ","End":"02:08.345","Text":"What happens if k is 0?"},{"Start":"02:08.345 ","End":"02:16.245","Text":"If k is 0, then we get that x squared plus y squared equals 1."},{"Start":"02:16.245 ","End":"02:17.545","Text":"We\u0027ve already done that."},{"Start":"02:17.545 ","End":"02:19.490","Text":"That\u0027s this circle here."},{"Start":"02:19.490 ","End":"02:22.605","Text":"I\u0027ll make another separate graph."},{"Start":"02:22.605 ","End":"02:25.845","Text":"I have a room for it here. That\u0027s the k equals 0."},{"Start":"02:25.845 ","End":"02:27.839","Text":"If k equals 1,"},{"Start":"02:27.839 ","End":"02:34.760","Text":"I\u0027ll get x squared plus y squared equals 0,"},{"Start":"02:34.760 ","End":"02:36.395","Text":"because 1 minus 1 squared."},{"Start":"02:36.395 ","End":"02:38.910","Text":"That will just be a point."},{"Start":"02:40.630 ","End":"02:46.070","Text":"If we put in some other values of k between 0 and 1,"},{"Start":"02:46.200 ","End":"02:51.220","Text":"we\u0027re just going to get circles between these 2."},{"Start":"02:51.220 ","End":"02:56.180","Text":"Another way to look at it is I could say 1 minus k squared is r squared,"},{"Start":"02:56.180 ","End":"03:00.630","Text":"where the radius is r. Because k is between 0 and 1,"},{"Start":"03:00.630 ","End":"03:04.165","Text":"1 minus k squared goes from 1 down to 0."},{"Start":"03:04.165 ","End":"03:11.910","Text":"We can say this is r squared where r is between 0 and 1."},{"Start":"03:11.910 ","End":"03:15.750","Text":"Here is radius 0, here is radius 1 and anything in between."},{"Start":"03:15.750 ","End":"03:17.640","Text":"Let\u0027s even draw another 1,"},{"Start":"03:17.640 ","End":"03:20.324","Text":"not the greatest circle."},{"Start":"03:20.324 ","End":"03:26.415","Text":"Anyway, you get the idea even if these circles are not precise."},{"Start":"03:26.415 ","End":"03:29.170","Text":"Let\u0027s leave it at that."}],"ID":8891},{"Watched":false,"Name":"Exercise 1 part e","Duration":"4m 1s","ChapterTopicVideoID":8518,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:02.940","Text":"This time we have a function of 2 variables,"},{"Start":"00:02.940 ","End":"00:04.710","Text":"natural log of x squared minus y."},{"Start":"00:04.710 ","End":"00:07.830","Text":"As usual, we have to define the domain and to sketch it,"},{"Start":"00:07.830 ","End":"00:11.190","Text":"and also to sketch the level curves."},{"Start":"00:11.190 ","End":"00:13.815","Text":"Let\u0027s first of all look at the domain."},{"Start":"00:13.815 ","End":"00:18.945","Text":"For the domain, the only problem is the natural logarithm."},{"Start":"00:18.945 ","End":"00:22.800","Text":"The argument of the natural log has to be bigger than 0."},{"Start":"00:22.800 ","End":"00:24.479","Text":"We get the equation,"},{"Start":"00:24.479 ","End":"00:27.570","Text":"x squared minus y bigger than 0."},{"Start":"00:27.570 ","End":"00:32.880","Text":"We have to see how that translates to sketch on the graph."},{"Start":"00:32.880 ","End":"00:34.665","Text":"If we look at it differently,"},{"Start":"00:34.665 ","End":"00:40.160","Text":"we get same thing as y is less than x squared."},{"Start":"00:40.160 ","End":"00:48.040","Text":"What we do is we sketch y equals x squared just to help us as a guideline."},{"Start":"00:48.040 ","End":"00:51.780","Text":"Let\u0027s see, x squared goes through 0, 0."},{"Start":"00:51.780 ","End":"00:58.110","Text":"If this is 1 and this is 1 minus 1, 1, 2, 4."},{"Start":"00:58.110 ","End":"01:00.365","Text":"I\u0027m just approximating."},{"Start":"01:00.365 ","End":"01:06.240","Text":"Anyway, we\u0027ll get something like this."},{"Start":"01:06.240 ","End":"01:08.535","Text":"Not so great. I\u0027ll try again."},{"Start":"01:08.535 ","End":"01:11.300","Text":"Still not the greatest, but it will do."},{"Start":"01:11.300 ","End":"01:12.920","Text":"This is y equals x squared."},{"Start":"01:12.920 ","End":"01:15.290","Text":"This is not in the domain."},{"Start":"01:15.290 ","End":"01:20.060","Text":"We need y strictly less than x squared and y decreases downwards."},{"Start":"01:20.060 ","End":"01:28.860","Text":"What we want is everything below but not including the parabola, strictly below."},{"Start":"01:29.450 ","End":"01:36.735","Text":"Basically everything below the parabola."},{"Start":"01:36.735 ","End":"01:44.160","Text":"This part inside the parabola is not in the domain. Rough sketch."},{"Start":"01:44.160 ","End":"01:50.135","Text":"Now, level curve is characterized by f of x,"},{"Start":"01:50.135 ","End":"01:52.475","Text":"y equals k. In other words,"},{"Start":"01:52.475 ","End":"01:56.000","Text":"natural log of x squared minus"},{"Start":"01:56.000 ","End":"02:04.175","Text":"y equals k. If we interpret the natural log in terms of the exponent,"},{"Start":"02:04.175 ","End":"02:12.540","Text":"we get that e^k equals x squared minus y,"},{"Start":"02:12.540 ","End":"02:14.480","Text":"and if we just change sides,"},{"Start":"02:14.480 ","End":"02:22.710","Text":"we get y equals x squared minus e^k."},{"Start":"02:22.710 ","End":"02:27.480","Text":"Now, as k runs along all possible numbers,"},{"Start":"02:27.700 ","End":"02:33.060","Text":"e^k is any positive number,"},{"Start":"02:33.580 ","End":"02:37.520","Text":"because e^k gives us all possible positive numbers."},{"Start":"02:37.520 ","End":"02:40.820","Text":"Any positive number we can take the log of it and that will be the"},{"Start":"02:40.820 ","End":"02:47.495","Text":"k. We could say that if we let e^k be c,"},{"Start":"02:47.495 ","End":"02:49.280","Text":"which is bigger than 0,"},{"Start":"02:49.280 ","End":"02:55.055","Text":"then we\u0027ve covered all possible k. All we have to do is for each c,"},{"Start":"02:55.055 ","End":"02:59.000","Text":"draw the graph of y equals x squared minus c."},{"Start":"02:59.000 ","End":"03:08.450","Text":"Let me just push this out of the way so I can get room for a graph."},{"Start":"03:08.450 ","End":"03:11.660","Text":"I started out with the same picture here."},{"Start":"03:11.660 ","End":"03:13.520","Text":"This is y equals x squared,"},{"Start":"03:13.520 ","End":"03:17.150","Text":"but this is not a level curve because I have to subtract some positive"},{"Start":"03:17.150 ","End":"03:21.410","Text":"c. Actually I have to be in this light green area."},{"Start":"03:21.410 ","End":"03:23.960","Text":"Let me just test, subtract various values,"},{"Start":"03:23.960 ","End":"03:26.555","Text":"which means pushing it down."},{"Start":"03:26.555 ","End":"03:30.590","Text":"Here\u0027s 1 example where I took some c and pushed it down."},{"Start":"03:30.590 ","End":"03:32.270","Text":"Let me draw a few others."},{"Start":"03:32.270 ","End":"03:34.050","Text":"Here\u0027s a couple more."},{"Start":"03:34.050 ","End":"03:36.170","Text":"They go further and further down."},{"Start":"03:36.170 ","End":"03:37.430","Text":"Just because of my drawing,"},{"Start":"03:37.430 ","End":"03:45.095","Text":"you don\u0027t see that they actually extend more rightwards and upwards."},{"Start":"03:45.095 ","End":"03:48.980","Text":"We drew them a bit better. We would see they would extend further here"},{"Start":"03:48.980 ","End":"03:52.805","Text":"and here and actually fill up the whole green area."},{"Start":"03:52.805 ","End":"03:54.125","Text":"Just bad sketching."},{"Start":"03:54.125 ","End":"03:56.730","Text":"But anyway you get the idea."},{"Start":"03:57.650 ","End":"04:01.570","Text":"Let\u0027s just call it quits at that."}],"ID":8892},{"Watched":false,"Name":"Exercise 1 part f","Duration":"4m 35s","ChapterTopicVideoID":8519,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"Here we\u0027re given the function f of x,"},{"Start":"00:02.970 ","End":"00:04.740","Text":"y is x square root of y,"},{"Start":"00:04.740 ","End":"00:07.230","Text":"we have to find the domain including a sketch,"},{"Start":"00:07.230 ","End":"00:09.465","Text":"and to sketch the level curves."},{"Start":"00:09.465 ","End":"00:11.685","Text":"Let\u0027s start with the domain."},{"Start":"00:11.685 ","End":"00:15.210","Text":"Now, let\u0027s look at what\u0027s not in the domain,"},{"Start":"00:15.210 ","End":"00:16.575","Text":"where is the problem?"},{"Start":"00:16.575 ","End":"00:18.990","Text":"The problem could be that y is negative,"},{"Start":"00:18.990 ","End":"00:20.985","Text":"and then it wouldn\u0027t be defined."},{"Start":"00:20.985 ","End":"00:25.055","Text":"As long as y is bigger or equal to 0,"},{"Start":"00:25.055 ","End":"00:28.120","Text":"we can do the square root and we can certainly multiply numbers,"},{"Start":"00:28.120 ","End":"00:29.945","Text":"so this is the only restriction."},{"Start":"00:29.945 ","End":"00:32.930","Text":"Now, what does y bigger or equal to 0 mean?"},{"Start":"00:32.930 ","End":"00:38.810","Text":"The positive y means above and including the x-axis,"},{"Start":"00:38.810 ","End":"00:42.590","Text":"it\u0027s the upper half-plane including the line."},{"Start":"00:42.590 ","End":"00:44.835","Text":"Let\u0027s sketch that."},{"Start":"00:44.835 ","End":"00:49.980","Text":"I\u0027ll draw some lines to indicate,"},{"Start":"00:49.980 ","End":"00:52.340","Text":"and perhaps just to emphasize,"},{"Start":"00:52.340 ","End":"00:57.905","Text":"I\u0027ll also highlight the x-axis that\u0027s part of it too,"},{"Start":"00:57.905 ","End":"00:59.735","Text":"up to and including."},{"Start":"00:59.735 ","End":"01:02.570","Text":"That\u0027s the domain."},{"Start":"01:02.570 ","End":"01:06.785","Text":"Next, let\u0027s see what we can say about level curves."},{"Start":"01:06.785 ","End":"01:09.245","Text":"Now, what is a level curve?"},{"Start":"01:09.245 ","End":"01:11.450","Text":"That would be to take f of x,"},{"Start":"01:11.450 ","End":"01:15.830","Text":"y equals k, in general, and in our case,"},{"Start":"01:15.830 ","End":"01:19.250","Text":"it would mean that x square root of y equals"},{"Start":"01:19.250 ","End":"01:24.320","Text":"k. I\u0027d like to try and get it as y in terms of x."},{"Start":"01:24.320 ","End":"01:28.780","Text":"Let\u0027s say we bring x to the other side."},{"Start":"01:28.780 ","End":"01:31.860","Text":"Well, let\u0027s assume that x is not 0."},{"Start":"01:31.860 ","End":"01:35.515","Text":"Let\u0027s, first of all, start off with k not being 0,"},{"Start":"01:35.515 ","End":"01:39.110","Text":"and then we\u0027ll take care of the k equals 0, separately."},{"Start":"01:39.110 ","End":"01:41.840","Text":"If k is not 0, then certainly x is not 0."},{"Start":"01:41.840 ","End":"01:48.520","Text":"So we can get that the square root of y is k/x,"},{"Start":"01:48.520 ","End":"01:57.030","Text":"and from here, I can get that y is equal to k squared over x squared."},{"Start":"01:57.620 ","End":"02:01.475","Text":"I\u0027ll put another pair of axes in here."},{"Start":"02:01.475 ","End":"02:03.080","Text":"Let\u0027s take some examples."},{"Start":"02:03.080 ","End":"02:05.060","Text":"Suppose I take k equals 1,"},{"Start":"02:05.060 ","End":"02:08.330","Text":"that will give me y equals 1/x squared."},{"Start":"02:08.330 ","End":"02:10.625","Text":"If I take k equals 2,"},{"Start":"02:10.625 ","End":"02:15.275","Text":"I\u0027ll get y equals 4/x squared and so on."},{"Start":"02:15.275 ","End":"02:20.800","Text":"1/x squared looks something like this."},{"Start":"02:20.800 ","End":"02:24.230","Text":"It\u0027s symmetrical about the y-axis,"},{"Start":"02:24.230 ","End":"02:27.455","Text":"it\u0027s an even function."},{"Start":"02:27.455 ","End":"02:30.570","Text":"When x is 1, y is 1."},{"Start":"02:30.950 ","End":"02:36.105","Text":"If k is 2, that\u0027s 4/x squared."},{"Start":"02:36.105 ","End":"02:38.340","Text":"Let\u0027s see if x is 1,"},{"Start":"02:38.340 ","End":"02:41.310","Text":"y is 4, it\u0027s similar,"},{"Start":"02:41.310 ","End":"02:43.990","Text":"it\u0027s just higher up."},{"Start":"02:44.570 ","End":"02:48.180","Text":"Same here and so on."},{"Start":"02:48.180 ","End":"02:54.435","Text":"When k is 3, we might get it\u0027s still higher, and so on."},{"Start":"02:54.435 ","End":"02:58.270","Text":"If k is less than 1 we\u0027ll get something like this,"},{"Start":"02:58.270 ","End":"03:02.640","Text":"and will fill almost the whole upper half-plane."},{"Start":"03:02.640 ","End":"03:04.910","Text":"Now, there\u0027s 1 thing we didn\u0027t take care of yet."},{"Start":"03:04.910 ","End":"03:08.110","Text":"What happens if k equals 0?"},{"Start":"03:08.110 ","End":"03:11.565","Text":"Well, in that case, we get a strange equation,"},{"Start":"03:11.565 ","End":"03:15.050","Text":"x times square root of y equals 0."},{"Start":"03:15.050 ","End":"03:18.755","Text":"Now, how do I sketch that?"},{"Start":"03:18.755 ","End":"03:20.750","Text":"Let\u0027s see what could be."},{"Start":"03:20.750 ","End":"03:24.605","Text":"The product of 2 numbers is 0 means that at least 1 of them is 0."},{"Start":"03:24.605 ","End":"03:28.800","Text":"If x is 0, then x is 0,"},{"Start":"03:28.800 ","End":"03:33.200","Text":"or if square root of y is 0,"},{"Start":"03:33.200 ","End":"03:38.750","Text":"then it means also that y equals 0."},{"Start":"03:38.750 ","End":"03:45.590","Text":"But we still have to stick to the restrictions,"},{"Start":"03:45.590 ","End":"03:47.645","Text":"so if x is 0,"},{"Start":"03:47.645 ","End":"03:52.050","Text":"y is still got to be bigger or equal to 0."},{"Start":"03:52.050 ","End":"03:53.870","Text":"I\u0027ll just note that if x is 0,"},{"Start":"03:53.870 ","End":"03:56.420","Text":"we still have y bigger or equal to 0."},{"Start":"03:56.420 ","End":"03:59.405","Text":"If y equals 0, there\u0027s no restriction on x."},{"Start":"03:59.405 ","End":"04:01.789","Text":"In other words, let\u0027s take the 2 bits separately,"},{"Start":"04:01.789 ","End":"04:03.570","Text":"this bit and this bit."},{"Start":"04:03.570 ","End":"04:07.430","Text":"X 0, y bigger or equal to 0 gives us"},{"Start":"04:07.430 ","End":"04:14.220","Text":"this part of the y-axis up to and including the origin."},{"Start":"04:15.830 ","End":"04:19.759","Text":"If y is 0 and x is unrestricted,"},{"Start":"04:19.759 ","End":"04:23.910","Text":"it gives us all of the x-axis."},{"Start":"04:23.910 ","End":"04:25.970","Text":"We have an upside-down T-shape,"},{"Start":"04:25.970 ","End":"04:30.320","Text":"a very unusual shape for the level curve for k equals 0."},{"Start":"04:30.320 ","End":"04:35.190","Text":"It could happen. Let\u0027s say that we\u0027re done."}],"ID":8893},{"Watched":false,"Name":"Exercise 1 part g","Duration":"4m ","ChapterTopicVideoID":8520,"CourseChapterTopicPlaylistID":4970,"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 a function of 3 variables,"},{"Start":"00:04.050 ","End":"00:08.460","Text":"f of x, y, z, which is x squared plus y squared plus z squared."},{"Start":"00:08.460 ","End":"00:17.880","Text":"We have to find what is the domain and sketch it or maybe not sketch it,"},{"Start":"00:17.880 ","End":"00:22.905","Text":"maybe just describe it because in 3 dimensions it\u0027s going to be a bit hard."},{"Start":"00:22.905 ","End":"00:25.650","Text":"To describe its level surfaces again,"},{"Start":"00:25.650 ","End":"00:27.810","Text":"it\u0027s pretty hard to sketch 3D,"},{"Start":"00:27.810 ","End":"00:31.035","Text":"so we\u0027ll settle for a description."},{"Start":"00:31.035 ","End":"00:34.080","Text":"Let\u0027s look at the domain first."},{"Start":"00:34.080 ","End":"00:36.620","Text":"What is the domain of f?"},{"Start":"00:36.620 ","End":"00:38.810","Text":"Well, are there any restrictions on x, y, and z?"},{"Start":"00:38.810 ","End":"00:39.950","Text":"Not really any x, y,"},{"Start":"00:39.950 ","End":"00:42.710","Text":"and z we can substitute here will be fine."},{"Start":"00:42.710 ","End":"00:44.810","Text":"It\u0027s all of x, y, z."},{"Start":"00:44.810 ","End":"00:51.570","Text":"Let\u0027s just say the whole 3D space."},{"Start":"00:51.570 ","End":"00:55.610","Text":"There is 1 way to describe it."},{"Start":"00:55.610 ","End":"00:58.265","Text":"No restriction on x, y, and z."},{"Start":"00:58.265 ","End":"01:01.189","Text":"Now let\u0027s look at the level curves."},{"Start":"01:01.189 ","End":"01:06.500","Text":"Level curves, remember, is when we take the function and say f of x,"},{"Start":"01:06.500 ","End":"01:08.300","Text":"y, z in this case,"},{"Start":"01:08.300 ","End":"01:14.420","Text":"and let it equal k. Now what we get is x squared plus y"},{"Start":"01:14.420 ","End":"01:21.260","Text":"squared plus z squared equals k. Now,"},{"Start":"01:21.260 ","End":"01:26.559","Text":"obviously, k is going to be bigger or equal to 0."},{"Start":"01:26.559 ","End":"01:31.055","Text":"The reason is because we have non-negative numbers here."},{"Start":"01:31.055 ","End":"01:32.770","Text":"If we add non-negative,"},{"Start":"01:32.770 ","End":"01:34.090","Text":"we have to have non-negative."},{"Start":"01:34.090 ","End":"01:37.990","Text":"We can\u0027t suddenly get a negative if we add zeros or positives."},{"Start":"01:37.990 ","End":"01:44.130","Text":"This is one restriction on k. Otherwise, we\u0027ll get, nothing."},{"Start":"01:44.130 ","End":"01:45.640","Text":"We won\u0027t get a level curve."},{"Start":"01:45.640 ","End":"01:47.770","Text":"But if k is bigger or equal to 0,"},{"Start":"01:47.770 ","End":"01:51.475","Text":"I can write k as r squared,"},{"Start":"01:51.475 ","End":"01:54.910","Text":"where r is, well, also non-negative."},{"Start":"01:54.910 ","End":"01:59.470","Text":"The reason I\u0027m doing this is because I recognize this as the equation of a sphere."},{"Start":"01:59.470 ","End":"02:06.380","Text":"I know that x squared plus y squared plus z squared equals r squared is a sphere."},{"Start":"02:06.380 ","End":"02:09.230","Text":"Actually you got to distinguish between 2 cases."},{"Start":"02:09.230 ","End":"02:13.200","Text":"Again, here r is bigger or equal to 0."},{"Start":"02:13.730 ","End":"02:15.785","Text":"I don\u0027t want it to be negative."},{"Start":"02:15.785 ","End":"02:17.690","Text":"Theoretically, I could take a negative radius,"},{"Start":"02:17.690 ","End":"02:20.270","Text":"but I might as well take the positive."},{"Start":"02:20.270 ","End":"02:22.745","Text":"If r equals 0,"},{"Start":"02:22.745 ","End":"02:24.455","Text":"we get one thing."},{"Start":"02:24.455 ","End":"02:27.780","Text":"Let\u0027s do that unusual 1 first."},{"Start":"02:29.600 ","End":"02:33.990","Text":"Let\u0027s separate, r equals 0 and r bigger than 0."},{"Start":"02:33.990 ","End":"02:35.690","Text":"First of all, r equals 0,"},{"Start":"02:35.690 ","End":"02:41.525","Text":"then we\u0027ll get x squared plus y squared plus z squared equals 0."},{"Start":"02:41.525 ","End":"02:44.135","Text":"That\u0027s just a single point."},{"Start":"02:44.135 ","End":"02:46.370","Text":"That\u0027s just the point x is 0,"},{"Start":"02:46.370 ","End":"02:48.530","Text":"y is 0 and z is 0."},{"Start":"02:48.530 ","End":"02:52.100","Text":"The only way I can add non-negative to get 0s if they\u0027re all 0,"},{"Start":"02:52.100 ","End":"02:54.650","Text":"if even 1 of them is positive, no good."},{"Start":"02:54.650 ","End":"02:57.300","Text":"That\u0027s a single point."},{"Start":"02:57.680 ","End":"03:03.795","Text":"Specifically, it\u0027s the origin in this case. That could happen."},{"Start":"03:03.795 ","End":"03:05.615","Text":"Level surface, I mean,"},{"Start":"03:05.615 ","End":"03:07.715","Text":"it\u0027s not really a surface, it\u0027s a point."},{"Start":"03:07.715 ","End":"03:10.234","Text":"But if r is bigger than 0,"},{"Start":"03:10.234 ","End":"03:13.580","Text":"then we get to x squared plus y"},{"Start":"03:13.580 ","End":"03:17.585","Text":"squared plus z squared equals r squared with a positive r,"},{"Start":"03:17.585 ","End":"03:23.060","Text":"then it\u0027s a sphere of radius"},{"Start":"03:23.060 ","End":"03:29.195","Text":"r. I\u0027m not going to sketch it."},{"Start":"03:29.195 ","End":"03:34.965","Text":"Also, you are expected to recognize in general the quadric surfaces."},{"Start":"03:34.965 ","End":"03:38.540","Text":"There\u0027s a chapter on quadric surfaces and one of them is a sphere,"},{"Start":"03:38.540 ","End":"03:40.955","Text":"and you should recognize it."},{"Start":"03:40.955 ","End":"03:47.494","Text":"That describes the level surfaces spheres around the origin."},{"Start":"03:47.494 ","End":"03:50.150","Text":"Well, I could also say they\u0027re centered at the origin."},{"Start":"03:50.150 ","End":"03:52.130","Text":"That\u0027s to be more precise."},{"Start":"03:52.130 ","End":"03:54.110","Text":"Centered at, let\u0027s say,"},{"Start":"03:54.110 ","End":"03:57.710","Text":"the origin 0 or the single point,"},{"Start":"03:57.710 ","End":"03:59.970","Text":"the origin. That\u0027s it."}],"ID":8894},{"Watched":false,"Name":"Exercise 1 part h","Duration":"5m 32s","ChapterTopicVideoID":8521,"CourseChapterTopicPlaylistID":4970,"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 a function of 3 variables, x, y,"},{"Start":"00:03.840 ","End":"00:05.280","Text":"and z, given as follows,"},{"Start":"00:05.280 ","End":"00:07.260","Text":"z squared minus x squared minus y squared."},{"Start":"00:07.260 ","End":"00:11.015","Text":"We have to find its domain of definition,"},{"Start":"00:11.015 ","End":"00:15.340","Text":"possibly not sketch it more like describe it,"},{"Start":"00:15.340 ","End":"00:18.360","Text":"because it\u0027s going to be a bit difficult in 3D"},{"Start":"00:18.360 ","End":"00:21.520","Text":"and to describe the level surfaces."},{"Start":"00:22.640 ","End":"00:26.475","Text":"Let\u0027s deal with the domain first."},{"Start":"00:26.475 ","End":"00:29.130","Text":"Is there any restriction on x,"},{"Start":"00:29.130 ","End":"00:32.520","Text":"y, or z, or this formula?"},{"Start":"00:32.520 ","End":"00:33.900","Text":"There\u0027s no restriction at all."},{"Start":"00:33.900 ","End":"00:37.395","Text":"Any x, y, and z I can compute this expression for."},{"Start":"00:37.395 ","End":"00:41.410","Text":"It\u0027s all x, y, and z."},{"Start":"00:41.780 ","End":"00:44.600","Text":"If I want to put it in other words,"},{"Start":"00:44.600 ","End":"00:47.090","Text":"it\u0027s the whole space,"},{"Start":"00:47.090 ","End":"00:50.450","Text":"the whole 3D space more precisely,"},{"Start":"00:50.450 ","End":"00:52.310","Text":"its actually 4D space,"},{"Start":"00:52.310 ","End":"00:55.470","Text":"5D space any number of dimensions."},{"Start":"00:55.840 ","End":"01:00.170","Text":"That\u0027s the domain and now level curves."},{"Start":"01:00.170 ","End":"01:03.920","Text":"If you remember, level curves are what happens"},{"Start":"01:03.920 ","End":"01:05.360","Text":"when we take f of x,"},{"Start":"01:05.360 ","End":"01:09.109","Text":"y, and z and let them equal some constant,"},{"Start":"01:09.109 ","End":"01:12.980","Text":"say k. In this case,"},{"Start":"01:12.980 ","End":"01:19.530","Text":"we will get z squared minus x squared minus y squared equals"},{"Start":"01:19.530 ","End":"01:24.000","Text":"k. Now it turns out that I get"},{"Start":"01:24.000 ","End":"01:26.690","Text":"different kinds of surfaces depending on"},{"Start":"01:26.690 ","End":"01:30.529","Text":"whether k is positive, negative, or 0."},{"Start":"01:30.529 ","End":"01:32.625","Text":"Let\u0027s take them. Let say,"},{"Start":"01:32.625 ","End":"01:35.385","Text":"first of all k equals 0,"},{"Start":"01:35.385 ","End":"01:42.420","Text":"then I\u0027ll take the case where k is bigger than 0,"},{"Start":"01:42.420 ","End":"01:48.170","Text":"and lastly we\u0027ll take the case where k is negative."},{"Start":"01:48.170 ","End":"01:50.615","Text":"Now, if k equals 0,"},{"Start":"01:50.615 ","End":"01:55.940","Text":"we get z squared minus x squared"},{"Start":"01:55.940 ","End":"02:00.880","Text":"minus y squared equals 0."},{"Start":"02:04.580 ","End":"02:08.710","Text":"If I bring the x squared and y squared to the other side,"},{"Start":"02:08.710 ","End":"02:13.090","Text":"I can say x squared plus y squared equals z squared."},{"Start":"02:13.090 ","End":"02:16.750","Text":"I\u0027d rather write it as x squared over 1 squared plus y"},{"Start":"02:16.750 ","End":"02:20.590","Text":"squared over 1 squared equals z squared over 1 squared."},{"Start":"02:20.590 ","End":"02:23.300","Text":"Why am I doing this strange thing?"},{"Start":"02:23.550 ","End":"02:27.475","Text":"I just copied the equation for"},{"Start":"02:27.475 ","End":"02:30.450","Text":"a cone that is centered on the z-axis"},{"Start":"02:30.450 ","End":"02:32.085","Text":"and this is the general form,"},{"Start":"02:32.085 ","End":"02:34.230","Text":"and here if I take a, b, c as 1,"},{"Start":"02:34.230 ","End":"02:36.195","Text":"I\u0027ll see that this is a cone."},{"Start":"02:36.195 ","End":"02:39.155","Text":"Let me just write the word cone."},{"Start":"02:39.155 ","End":"02:42.010","Text":"Specifically, it\u0027s a cone that opens up"},{"Start":"02:42.010 ","End":"02:46.555","Text":"in the z direction or centered on the z-axis."},{"Start":"02:46.555 ","End":"02:48.940","Text":"Now, if k is bigger than 0,"},{"Start":"02:48.940 ","End":"02:56.315","Text":"then I can write k as equal to some other constant squared,"},{"Start":"02:56.315 ","End":"03:01.105","Text":"where c is not 0, bigger than 0."},{"Start":"03:01.105 ","End":"03:07.090","Text":"Now this time I get that z squared minus x squared"},{"Start":"03:07.090 ","End":"03:09.340","Text":"minus y squared equals k,"},{"Start":"03:09.340 ","End":"03:10.929","Text":"which is c squared,"},{"Start":"03:10.929 ","End":"03:12.895","Text":"and if I rewrite this,"},{"Start":"03:12.895 ","End":"03:16.780","Text":"I can get it in the form if I divide by c squared,"},{"Start":"03:16.780 ","End":"03:19.945","Text":"and I also slightly change the order I can write,"},{"Start":"03:19.945 ","End":"03:27.444","Text":"minus x squared over c squared minus y squared"},{"Start":"03:27.444 ","End":"03:36.065","Text":"over c squared plus z squared over c squared equals 1."},{"Start":"03:36.065 ","End":"03:39.340","Text":"Let me show you a formula which I got"},{"Start":"03:39.340 ","End":"03:43.310","Text":"from the section on quadric surfaces,"},{"Start":"03:45.510 ","End":"03:51.930","Text":"and this is the general form of a hyperboloid."},{"Start":"03:51.930 ","End":"03:56.699","Text":"Again, which is in the direction of the z-axis,"},{"Start":"03:56.699 ","End":"04:00.780","Text":"the z is the exception variable here."},{"Start":"04:00.780 ","End":"04:02.805","Text":"There\u0027s 2 kinds of hyperboloid,"},{"Start":"04:02.805 ","End":"04:06.219","Text":"and this 1 is the 1 with 2 sheets."},{"Start":"04:06.219 ","End":"04:09.250","Text":"As you might guess, the next 1 will be a hyperboloid"},{"Start":"04:09.250 ","End":"04:11.610","Text":"with 1 sheet but let\u0027s continue."},{"Start":"04:11.610 ","End":"04:13.970","Text":"Now, if k is less than 0,"},{"Start":"04:13.970 ","End":"04:19.950","Text":"then I can say that k is equal to minus c squared,"},{"Start":"04:19.950 ","End":"04:23.795","Text":"where c is some number, positive."},{"Start":"04:23.795 ","End":"04:30.560","Text":"Then I get that z squared minus x squared minus y"},{"Start":"04:30.560 ","End":"04:34.280","Text":"squared equals minus c squared."},{"Start":"04:34.280 ","End":"04:36.770","Text":"I\u0027m going to need to change some order around here,"},{"Start":"04:36.770 ","End":"04:39.260","Text":"I\u0027m going to divide by minus c squared,"},{"Start":"04:39.260 ","End":"04:42.230","Text":"and I\u0027m also going to write it in the order of x, y, and z,"},{"Start":"04:42.230 ","End":"04:46.070","Text":"so what this will give us this time is it will give us,"},{"Start":"04:46.070 ","End":"04:48.800","Text":"just like the above but the sign is reversed,"},{"Start":"04:48.800 ","End":"04:53.690","Text":"x squared over c squared plus y squared over c"},{"Start":"04:53.690 ","End":"05:00.515","Text":"squared minus z squared over c squared equals 1."},{"Start":"05:00.515 ","End":"05:03.515","Text":"Again, if you refer to quadric sections,"},{"Start":"05:03.515 ","End":"05:08.310","Text":"this 1 will be also a hyperboloid,"},{"Start":"05:08.930 ","End":"05:14.170","Text":"but only 1 sheet"},{"Start":"05:17.110 ","End":"05:20.735","Text":"and more specifically centered around the z-axis."},{"Start":"05:20.735 ","End":"05:22.010","Text":"In all these cases,"},{"Start":"05:22.010 ","End":"05:24.560","Text":"the z is the exception variable."},{"Start":"05:24.560 ","End":"05:28.570","Text":"As you see, z is positive x and y are negative."},{"Start":"05:28.570 ","End":"05:32.490","Text":"That\u0027s all. We are done."}],"ID":8895},{"Watched":false,"Name":"Exercise 2 part a","Duration":"2m 50s","ChapterTopicVideoID":8540,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.575","Text":"In this exercise, we have to compute the limit,"},{"Start":"00:03.575 ","End":"00:07.005","Text":"but notice that we have a function of 2 variables,"},{"Start":"00:07.005 ","End":"00:12.390","Text":"x and y, so this is a double variable limit or whatever,"},{"Start":"00:12.390 ","End":"00:16.185","Text":"x,y goes to 0,0 the origin."},{"Start":"00:16.185 ","End":"00:20.460","Text":"Now, you might say, what\u0027s the problem, just to substitute?"},{"Start":"00:20.460 ","End":"00:23.625","Text":"The thing is that just substitution doesn\u0027t work."},{"Start":"00:23.625 ","End":"00:32.925","Text":"If we try to take sine of x cubed y over x cubed y,"},{"Start":"00:32.925 ","End":"00:37.919","Text":"and we try to substitute x equals 0, y equals 0,"},{"Start":"00:37.919 ","End":"00:46.430","Text":"then what we get is sine of x cubed y is 0 times 0 is 0 over 0."},{"Start":"00:46.430 ","End":"00:50.990","Text":"In short, we have a 0 over 0 situation,"},{"Start":"00:50.990 ","End":"00:53.700","Text":"so we can\u0027t just substitute."},{"Start":"00:53.700 ","End":"00:56.315","Text":"We\u0027re going to have to use some trick."},{"Start":"00:56.315 ","End":"00:59.255","Text":"Now, the first trick to use,"},{"Start":"00:59.255 ","End":"01:02.210","Text":"and this is something that occurs often,"},{"Start":"01:02.210 ","End":"01:06.290","Text":"is we can reduce it to a function of 1 variable."},{"Start":"01:06.290 ","End":"01:11.235","Text":"Notice that we have x cubed y over x cubed y here,"},{"Start":"01:11.235 ","End":"01:14.550","Text":"so if we let some variable, call it t,"},{"Start":"01:14.550 ","End":"01:18.630","Text":"equals x cubed y,"},{"Start":"01:18.630 ","End":"01:21.750","Text":"then this limit becomes the limit"},{"Start":"01:21.750 ","End":"01:24.180","Text":"and I\u0027ll leave this blank for the moment,"},{"Start":"01:24.180 ","End":"01:29.925","Text":"of sine of t over t."},{"Start":"01:29.925 ","End":"01:37.950","Text":"Notice that when x goes to 0 and y goes to 0,"},{"Start":"01:37.950 ","End":"01:40.860","Text":"then t also goes to 0,"},{"Start":"01:40.860 ","End":"01:43.320","Text":"so we have t goes to 0."},{"Start":"01:43.320 ","End":"01:49.860","Text":"It\u0027s the same 0 we computed only that x cubed y here be 0, 0."},{"Start":"01:49.860 ","End":"01:55.360","Text":"Now we have a 1 variable problem."},{"Start":"01:55.360 ","End":"01:57.730","Text":"We still have 0 over 0,"},{"Start":"01:57.730 ","End":"02:01.390","Text":"but when we have functions of 1 variable in our bag of tricks,"},{"Start":"02:01.390 ","End":"02:02.800","Text":"we have L\u0027Hospital\u0027s rule."},{"Start":"02:02.800 ","End":"02:04.060","Text":"Let me just write his name."},{"Start":"02:04.060 ","End":"02:07.470","Text":"It\u0027s a French name, spelled funny."},{"Start":"02:07.470 ","End":"02:10.830","Text":"Well, not funny to the French, but L\u0027Hopital."},{"Start":"02:10.830 ","End":"02:15.335","Text":"Which says that if you have a 0 over 0 situation,"},{"Start":"02:15.335 ","End":"02:18.095","Text":"you can instead of computing this limit,"},{"Start":"02:18.095 ","End":"02:20.840","Text":"compute a different limit,"},{"Start":"02:20.840 ","End":"02:26.000","Text":"which is what you get if you differentiate numerator and denominator separately."},{"Start":"02:26.000 ","End":"02:29.960","Text":"In the numerator, we get cosine t,"},{"Start":"02:29.960 ","End":"02:33.190","Text":"and in the denominator, we get 1."},{"Start":"02:33.190 ","End":"02:36.300","Text":"Now we can easily let t go to 0,"},{"Start":"02:36.300 ","End":"02:38.115","Text":"cosine of 0 is 1,"},{"Start":"02:38.115 ","End":"02:39.900","Text":"we get 1 over 1,"},{"Start":"02:39.900 ","End":"02:41.910","Text":"and the answer is 1."},{"Start":"02:41.910 ","End":"02:46.040","Text":"This is therefore the answer to the original question,"},{"Start":"02:46.040 ","End":"02:49.980","Text":"so I\u0027ll just highlight it and declare that we are done."}],"ID":8896},{"Watched":false,"Name":"Exercise 2 part b","Duration":"2m 49s","ChapterTopicVideoID":8541,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this exercise, we have to compute a limit."},{"Start":"00:02.760 ","End":"00:05.670","Text":"Again, we have functions of 2 variables,"},{"Start":"00:05.670 ","End":"00:07.755","Text":"so we have to substitute x and y."},{"Start":"00:07.755 ","End":"00:12.240","Text":"This time the limit is at 3,2."},{"Start":"00:12.240 ","End":"00:16.830","Text":"The reason that it\u0027s a problem is that we can\u0027t do straightforward substitution,"},{"Start":"00:16.830 ","End":"00:18.690","Text":"because if we do substitution,"},{"Start":"00:18.690 ","End":"00:21.735","Text":"we get sine of 3 times 2."},{"Start":"00:21.735 ","End":"00:27.135","Text":"I mean, if we substitute x equals 3, y equals 2,"},{"Start":"00:27.135 ","End":"00:36.000","Text":"then we\u0027d get sine of 3 times 2 minus 6 over 3 squared 2 squared minus 36."},{"Start":"00:36.000 ","End":"00:37.110","Text":"Now, if you look at it,"},{"Start":"00:37.110 ","End":"00:40.305","Text":"this is sine of 0 and this is,"},{"Start":"00:40.305 ","End":"00:42.045","Text":"9 times 4 is 36,"},{"Start":"00:42.045 ","End":"00:45.255","Text":"which basically becomes 0 over 0,"},{"Start":"00:45.255 ","End":"00:47.579","Text":"so there\u0027s a problem."},{"Start":"00:47.579 ","End":"00:51.780","Text":"There\u0027s no L\u0027Hospital in functions of 2 variables,"},{"Start":"00:51.780 ","End":"00:54.545","Text":"so we\u0027ll use a trick we\u0027ve used before."},{"Start":"00:54.545 ","End":"00:59.665","Text":"We can see that if we let t equals xy,"},{"Start":"00:59.665 ","End":"01:02.600","Text":"then we express everything in terms of t."},{"Start":"01:02.600 ","End":"01:05.704","Text":"What we can get is the limit,"},{"Start":"01:05.704 ","End":"01:07.640","Text":"leave this for a moment,"},{"Start":"01:07.640 ","End":"01:18.325","Text":"of the sine of xy is t minus 6 over,"},{"Start":"01:18.325 ","End":"01:22.320","Text":"now x squared y squared is the same as xy all squared,"},{"Start":"01:22.320 ","End":"01:23.460","Text":"so that\u0027s t squared,"},{"Start":"01:23.460 ","End":"01:27.320","Text":"just using the rules of exponents, minus 36."},{"Start":"01:27.320 ","End":"01:32.765","Text":"Now, when x goes to 3 and y goes to 2,"},{"Start":"01:32.765 ","End":"01:39.165","Text":"then xy turns to 3 times 2 which is 6,"},{"Start":"01:39.165 ","End":"01:43.935","Text":"and that\u0027s t, so t here turns to 6."},{"Start":"01:43.935 ","End":"01:47.180","Text":"Now, we still have the 0 over 0 because"},{"Start":"01:47.180 ","End":"01:51.425","Text":"t minus 6 is 0 and t squared minus 36 is also 0."},{"Start":"01:51.425 ","End":"01:52.975","Text":"But in this case,"},{"Start":"01:52.975 ","End":"01:55.170","Text":"when the function of 1 variable t,"},{"Start":"01:55.170 ","End":"01:57.360","Text":"we can use L\u0027Hopital\u0027s rule."},{"Start":"01:57.360 ","End":"02:01.030","Text":"I like to write his name just to give him credit."},{"Start":"02:01.070 ","End":"02:05.340","Text":"According to L\u0027Hopital when we have a 0 over 0 situation,"},{"Start":"02:05.340 ","End":"02:08.645","Text":"we can replace this limit by a different limit,"},{"Start":"02:08.645 ","End":"02:13.475","Text":"which is what we get when we differentiate the numerator separately."},{"Start":"02:13.475 ","End":"02:23.940","Text":"That\u0027s cosine of t minus 6 times inner derivative is 1 over derivative here is 2t."},{"Start":"02:24.470 ","End":"02:29.885","Text":"Now, there\u0027s no problem here in substituting t equals 6."},{"Start":"02:29.885 ","End":"02:42.570","Text":"Basically what I get is cosine of 0 over 2 times t is 2 times 6, is 12, cosine 0 is 1."},{"Start":"02:42.570 ","End":"02:44.895","Text":"The answer is 1/12."},{"Start":"02:44.895 ","End":"02:49.270","Text":"I\u0027ll just highlight it and we\u0027re done."}],"ID":8897},{"Watched":false,"Name":"Exercise 2 part c","Duration":"4m 4s","ChapterTopicVideoID":8542,"CourseChapterTopicPlaylistID":4970,"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 another limit,"},{"Start":"00:03.150 ","End":"00:06.285","Text":"also 2 variables, x, and y."},{"Start":"00:06.285 ","End":"00:08.700","Text":"Limit as x, y goes to 1,"},{"Start":"00:08.700 ","End":"00:12.060","Text":"2 of this expression."},{"Start":"00:12.060 ","End":"00:19.065","Text":"Notice that the problem is that we can\u0027t just substitute, x, y equals 1, 2,"},{"Start":"00:19.065 ","End":"00:25.620","Text":"because then we would get the arctangent of 1 plus"},{"Start":"00:25.620 ","End":"00:33.930","Text":"2 minus 3 over natural log of 1 plus 2 minus 2."},{"Start":"00:33.930 ","End":"00:36.900","Text":"Now, this is arctangent of,"},{"Start":"00:36.900 ","End":"00:38.970","Text":"this bit here is 0,"},{"Start":"00:38.970 ","End":"00:40.755","Text":"this bit here is 1,"},{"Start":"00:40.755 ","End":"00:47.810","Text":"arctangent of 0 is 0 and natural log of 1 is also 0."},{"Start":"00:47.810 ","End":"00:50.105","Text":"So we have a 0 over 0."},{"Start":"00:50.105 ","End":"00:56.175","Text":"But there\u0027s no L\u0027Hopital for functions of 2 variables in the limits."},{"Start":"00:56.175 ","End":"00:58.575","Text":"So we need the standard trick,"},{"Start":"00:58.575 ","End":"01:06.305","Text":"is to replace some expression in x and y as the variable t. In this case,"},{"Start":"01:06.305 ","End":"01:08.405","Text":"x plus y is the common bit,"},{"Start":"01:08.405 ","End":"01:14.560","Text":"so if I let t equal x plus y, the substitution."},{"Start":"01:14.560 ","End":"01:16.500","Text":"Then notice that when x,"},{"Start":"01:16.500 ","End":"01:18.315","Text":"y goes to 1, 2,"},{"Start":"01:18.315 ","End":"01:23.774","Text":"then t goes to 1 plus 2, which is 3,"},{"Start":"01:23.774 ","End":"01:26.390","Text":"and so when we do this substitution,"},{"Start":"01:26.390 ","End":"01:35.285","Text":"we get the limit as t goes to 3 of arctangent of"},{"Start":"01:35.285 ","End":"01:40.715","Text":"t minus 3 over"},{"Start":"01:40.715 ","End":"01:46.070","Text":"natural logarithm of t minus 2."},{"Start":"01:46.070 ","End":"01:48.200","Text":"Of course, we have the same problem as before,"},{"Start":"01:48.200 ","End":"01:50.210","Text":"that if we let t equal 3,"},{"Start":"01:50.210 ","End":"01:54.500","Text":"we get arctangent of 0 over natural log of 1,"},{"Start":"01:54.500 ","End":"01:56.090","Text":"0 over 0 again."},{"Start":"01:56.090 ","End":"01:58.955","Text":"But now we\u0027re in the 1 variable case."},{"Start":"01:58.955 ","End":"02:02.405","Text":"So we can use L\u0027Hopital\u0027s rule,"},{"Start":"02:02.405 ","End":"02:05.840","Text":"which says that we can differentiate numerator and"},{"Start":"02:05.840 ","End":"02:09.640","Text":"denominator separately and get a new limit which is equal,"},{"Start":"02:09.640 ","End":"02:13.700","Text":"so we get the limit as t goes to 3."},{"Start":"02:13.700 ","End":"02:16.080","Text":"Let\u0027s just remember that,"},{"Start":"02:18.280 ","End":"02:23.480","Text":"if it was arctangent of x derivative,"},{"Start":"02:23.480 ","End":"02:25.950","Text":"it\u0027s 1 over 1 plus x squared,"},{"Start":"02:25.950 ","End":"02:28.775","Text":"and just in case you\u0027ve drawn a blank,"},{"Start":"02:28.775 ","End":"02:33.340","Text":"the derivative of natural log is 1 over x."},{"Start":"02:33.340 ","End":"02:37.214","Text":"In this case, instead of x, we have t minus 3,"},{"Start":"02:37.214 ","End":"02:43.505","Text":"so we have 1 over 1 plus t minus 3"},{"Start":"02:43.505 ","End":"02:53.135","Text":"squared over 1 over t minus 2."},{"Start":"02:53.135 ","End":"02:57.755","Text":"Now notice that normally we would multiply by the inner derivative,"},{"Start":"02:57.755 ","End":"03:00.830","Text":"the derivative of t minus 3,"},{"Start":"03:00.830 ","End":"03:02.930","Text":"but that\u0027s just 1."},{"Start":"03:02.930 ","End":"03:05.960","Text":"The derivative of t minus 2 is also 1,"},{"Start":"03:05.960 ","End":"03:09.545","Text":"so we don\u0027t have to make any adjustments."},{"Start":"03:09.545 ","End":"03:12.965","Text":"Let\u0027s just figure out what this is."},{"Start":"03:12.965 ","End":"03:19.190","Text":"Well, we have a fraction over a fraction and when we have a fraction,"},{"Start":"03:19.190 ","End":"03:22.790","Text":"we divide fractions, we multiply by the inverse fraction."},{"Start":"03:22.790 ","End":"03:26.930","Text":"Basically what I\u0027m saying is that this thing comes down to the denominator,"},{"Start":"03:26.930 ","End":"03:30.470","Text":"1 plus t minus 3 squared,"},{"Start":"03:30.470 ","End":"03:34.345","Text":"and this thing goes up to the numerator t minus 2."},{"Start":"03:34.345 ","End":"03:39.135","Text":"Now let\u0027s see if we let t equals 3 here,"},{"Start":"03:39.135 ","End":"03:43.355","Text":"there\u0027s no problem, because on the numerator,"},{"Start":"03:43.355 ","End":"03:49.590","Text":"we get 3 minus 2 is 1,"},{"Start":"03:49.590 ","End":"03:51.570","Text":"and on the denominator,"},{"Start":"03:51.570 ","End":"03:55.100","Text":"3 minus 3 squared is 0,"},{"Start":"03:55.100 ","End":"03:56.810","Text":"so it\u0027s 1 over 1,"},{"Start":"03:56.810 ","End":"03:58.870","Text":"so it\u0027s just 1."},{"Start":"03:58.870 ","End":"04:05.440","Text":"No problem. I\u0027ll just highlight it because that\u0027s our answer, and we\u0027re done."}],"ID":8898},{"Watched":false,"Name":"Exercise 2 part d","Duration":"3m 16s","ChapterTopicVideoID":8543,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"In this exercise, we have the following limit to compute."},{"Start":"00:03.075 ","End":"00:07.369","Text":"Notice that the expression is a function of 2 variables,"},{"Start":"00:07.369 ","End":"00:10.920","Text":"x and y, and the limit is x goes to 0,"},{"Start":"00:10.920 ","End":"00:13.845","Text":"y goes to 0 from above."},{"Start":"00:13.845 ","End":"00:17.760","Text":"We\u0027re going to use our usual trick."},{"Start":"00:17.760 ","End":"00:22.020","Text":"I\u0027m not even going to discuss what\u0027s the problem, I\u0027m just substituting."},{"Start":"00:22.020 ","End":"00:27.509","Text":"It turns out that we get 0 times minus infinity,"},{"Start":"00:27.509 ","End":"00:29.910","Text":"but we can just, in any event,"},{"Start":"00:29.910 ","End":"00:33.540","Text":"substitute t is equal to x squared plus y,"},{"Start":"00:33.540 ","End":"00:34.650","Text":"because it appears twice."},{"Start":"00:34.650 ","End":"00:38.470","Text":"Let\u0027s let t equals x squared y."},{"Start":"00:38.470 ","End":"00:46.310","Text":"Then, when x,y, goes to 0,0 plus,"},{"Start":"00:46.310 ","End":"00:52.350","Text":"then x squared will go to 0 from above,"},{"Start":"00:52.350 ","End":"00:54.650","Text":"it\u0027s also going to be 0 plus,"},{"Start":"00:54.650 ","End":"01:00.200","Text":"and y goes to 0 plus 0 plus times 0 plus, is 0 plus."},{"Start":"01:00.200 ","End":"01:04.485","Text":"After substituting we get the limit,"},{"Start":"01:04.485 ","End":"01:08.325","Text":"as t goes to 0 plus,"},{"Start":"01:08.325 ","End":"01:13.715","Text":"meaning 0 from the right or from above of t,"},{"Start":"01:13.715 ","End":"01:19.950","Text":"natural log of t. Now,"},{"Start":"01:19.950 ","End":"01:26.535","Text":"this is 0 times minus infinity."},{"Start":"01:26.535 ","End":"01:29.690","Text":"There\u0027s a trick we use in order to use L\u0027Hopital."},{"Start":"01:29.690 ","End":"01:31.610","Text":"It\u0027s just an algebraic trick."},{"Start":"01:31.610 ","End":"01:36.665","Text":"In general, if we have a times b and then not 0,"},{"Start":"01:36.665 ","End":"01:43.650","Text":"then we can write this as b divided by 1 over a,"},{"Start":"01:43.650 ","End":"01:47.805","Text":"or vice versa, a over 1 over b, it doesn\u0027t matter."},{"Start":"01:47.805 ","End":"01:49.710","Text":"If we do that here,"},{"Start":"01:49.710 ","End":"01:51.719","Text":"then we get the limit,"},{"Start":"01:51.719 ","End":"01:56.100","Text":"as t goes to 0 plus of,"},{"Start":"01:56.100 ","End":"01:58.330","Text":"and I\u0027ll take this one into the denominator,"},{"Start":"01:58.330 ","End":"02:03.040","Text":"natural log of t divided by 1 over t."},{"Start":"02:03.040 ","End":"02:08.260","Text":"Now at this point, we have an infinity over infinity."},{"Start":"02:08.260 ","End":"02:13.770","Text":"Well, it\u0027s really minus infinity over infinity,"},{"Start":"02:13.770 ","End":"02:16.615","Text":"because when t goes to 0 plus this goes to plus infinity."},{"Start":"02:16.615 ","End":"02:18.865","Text":"I\u0027ll just emphasize it with a plus."},{"Start":"02:18.865 ","End":"02:24.010","Text":"L\u0027Hopital\u0027s rule works also for minus infinity over infinity."},{"Start":"02:24.010 ","End":"02:27.850","Text":"I just mentioned his name, L\u0027Hopital,"},{"Start":"02:27.850 ","End":"02:33.000","Text":"the French mathematician who discovered this property,"},{"Start":"02:33.000 ","End":"02:37.400","Text":"that we can replace this by the limit,"},{"Start":"02:37.400 ","End":"02:39.755","Text":"where t goes to the same thing,"},{"Start":"02:39.755 ","End":"02:42.470","Text":"but of the derivative of the top and bottom."},{"Start":"02:42.470 ","End":"02:45.860","Text":"Derivative of the top is 1 over t,"},{"Start":"02:45.860 ","End":"02:50.495","Text":"derivative of the bottom is minus 1 over t squared."},{"Start":"02:50.495 ","End":"02:52.805","Text":"Now this, if you compute it,"},{"Start":"02:52.805 ","End":"03:00.030","Text":"is just equal to minus t. Now,"},{"Start":"03:00.030 ","End":"03:05.250","Text":"the limit as t goes to 0 plus or minus t, is just 0."},{"Start":"03:05.250 ","End":"03:07.170","Text":"It doesn\u0027t matter if from above, from below,"},{"Start":"03:07.170 ","End":"03:10.325","Text":"I just can substitute t equals 0, we get 0."},{"Start":"03:10.325 ","End":"03:16.240","Text":"The answer is 0. I\u0027ll highlight it. We are done."}],"ID":8899},{"Watched":false,"Name":"Exercise 2 part e","Duration":"5m 12s","ChapterTopicVideoID":8544,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.910","Text":"In this exercise, we have another limit to compute in 2 variables, x and y."},{"Start":"00:06.950 ","End":"00:13.260","Text":"As usual, if you just substitute x is 1,"},{"Start":"00:13.260 ","End":"00:20.160","Text":"y is 1, you will get sine of 0 over 0, and it won\u0027t work,"},{"Start":"00:20.160 ","End":"00:22.409","Text":"so we have to use some tricks,"},{"Start":"00:22.409 ","End":"00:25.770","Text":"and the usual trick is a substitution."},{"Start":"00:25.770 ","End":"00:31.320","Text":"In this case, we see that x plus 2y minus 3 appears on both,"},{"Start":"00:31.320 ","End":"00:38.650","Text":"so we say, let t equal x plus 2y minus 3."},{"Start":"00:39.050 ","End":"00:44.975","Text":"That\u0027s not all, because we also have to see what t goes to."},{"Start":"00:44.975 ","End":"00:53.430","Text":"Now, when x, y goes to 1 from the right or from above, and y also,"},{"Start":"00:53.430 ","End":"00:56.190","Text":"they both go to 1 from above,"},{"Start":"00:56.190 ","End":"01:01.480","Text":"then t goes to"},{"Start":"01:02.120 ","End":"01:07.220","Text":"1 plus twice 1 minus 3 is 0,"},{"Start":"01:07.220 ","End":"01:09.530","Text":"but it\u0027s from above because this is a bit more than 1,"},{"Start":"01:09.530 ","End":"01:11.240","Text":"it\u0027ll be a bit more than 2,"},{"Start":"01:11.240 ","End":"01:13.490","Text":"and so it\u0027s just over 3."},{"Start":"01:13.490 ","End":"01:18.150","Text":"So t goes to 3 from above, from the right."},{"Start":"01:19.210 ","End":"01:29.420","Text":"What we get is the limit, as t goes to 3 from the right of"},{"Start":"01:29.420 ","End":"01:33.815","Text":"sine of square root of"},{"Start":"01:33.815 ","End":"01:40.815","Text":"t over t. Now,"},{"Start":"01:40.815 ","End":"01:45.560","Text":"the importance of the 3 from the right is that t"},{"Start":"01:45.560 ","End":"01:51.545","Text":"goes to 3 through values that are like 3.1, 3.01, 3.001,"},{"Start":"01:51.545 ","End":"02:01.530","Text":"whatever it is, t goes to,"},{"Start":"02:01.530 ","End":"02:04.679","Text":"and this is symbolic, 1 plus,"},{"Start":"02:04.679 ","End":"02:09.395","Text":"plus twice 1 plus minus 3."},{"Start":"02:09.395 ","End":"02:12.490","Text":"Now, this means a little bit above 1,"},{"Start":"02:12.490 ","End":"02:15.780","Text":"and twice that will be a little bit above 2,"},{"Start":"02:15.780 ","End":"02:19.715","Text":"so altogether we have a little bit over 3 minus 3,"},{"Start":"02:19.715 ","End":"02:23.415","Text":"so it will be just above 0."},{"Start":"02:23.415 ","End":"02:27.780","Text":"We can say that t goes to 0 plus."},{"Start":"02:27.780 ","End":"02:30.330","Text":"The importance of this,"},{"Start":"02:30.330 ","End":"02:35.090","Text":"the reason that t has to go to 0 from the right and be slightly positive is for"},{"Start":"02:35.090 ","End":"02:37.460","Text":"the square root to make sense, because"},{"Start":"02:37.460 ","End":"02:41.725","Text":"the square root won\u0027t make sense for negative values."},{"Start":"02:41.725 ","End":"02:47.480","Text":"What we get is the limit, as t goes to"},{"Start":"02:47.480 ","End":"02:54.735","Text":"0 from the right of sine of square root of t,"},{"Start":"02:54.735 ","End":"02:58.310","Text":"which as we said, is defined because t is going to be going to 0"},{"Start":"02:58.310 ","End":"03:03.440","Text":"through positive values over t. Now,"},{"Start":"03:03.440 ","End":"03:08.435","Text":"we still have a 0 over 0 situation, but in 1 variable,"},{"Start":"03:08.435 ","End":"03:12.185","Text":"so we can now apply L\u0027Hopital\u0027s Rule,"},{"Start":"03:12.185 ","End":"03:15.320","Text":"which says that in a 0 over 0 situation,"},{"Start":"03:15.320 ","End":"03:18.170","Text":"we can differentiate the top and the bottom separately and"},{"Start":"03:18.170 ","End":"03:21.275","Text":"get something equivalent, and the same answer."},{"Start":"03:21.275 ","End":"03:24.235","Text":"So t goes to 0 plus."},{"Start":"03:24.235 ","End":"03:27.940","Text":"Now, derivative of the denominator is 1,"},{"Start":"03:27.940 ","End":"03:31.085","Text":"so we can just forget about that or not,"},{"Start":"03:31.085 ","End":"03:32.915","Text":"I\u0027ll just write it just in case."},{"Start":"03:32.915 ","End":"03:36.180","Text":"Derivative of sine is cosine."},{"Start":"03:36.460 ","End":"03:39.890","Text":"But that\u0027s not the end of it, because it\u0027s not t,"},{"Start":"03:39.890 ","End":"03:41.090","Text":"it\u0027s an expression in t,"},{"Start":"03:41.090 ","End":"03:43.670","Text":"so we have to multiply by the derivative of that."},{"Start":"03:43.670 ","End":"03:51.460","Text":"Derivative of square root is 1 over twice the square root of t. Now,"},{"Start":"03:51.620 ","End":"03:56.715","Text":"when t goes to 0 from above, what we get,"},{"Start":"03:56.715 ","End":"03:57.840","Text":"just looking at the numerator,"},{"Start":"03:57.840 ","End":"04:02.369","Text":"cosine of 0, that\u0027s okay."},{"Start":"04:02.369 ","End":"04:06.015","Text":"Cosine square root of 0 is cosine of 0 is 1."},{"Start":"04:06.015 ","End":"04:12.465","Text":"This is 1 times 1 over 2 times,"},{"Start":"04:12.465 ","End":"04:16.275","Text":"and as t goes to 0 from above so does square root of t,"},{"Start":"04:16.275 ","End":"04:19.975","Text":"times 0 plus, I don\u0027t have to put over 1."},{"Start":"04:19.975 ","End":"04:28.250","Text":"Essentially, what I\u0027m getting is 1 over twice 0 plus."},{"Start":"04:28.250 ","End":"04:32.590","Text":"Now 0 plus in the denominator means it goes to plus infinity."},{"Start":"04:32.590 ","End":"04:37.279","Text":"Let\u0027s say that I have 1 times 1/2 times infinity,"},{"Start":"04:37.279 ","End":"04:38.945","Text":"which is just infinity,"},{"Start":"04:38.945 ","End":"04:41.735","Text":"this is plus infinity."},{"Start":"04:41.735 ","End":"04:48.515","Text":"It\u0027s a 1/2 times 1/0 plus,"},{"Start":"04:48.515 ","End":"04:50.900","Text":"and this tends to infinity."},{"Start":"04:50.900 ","End":"04:54.515","Text":"Something goes to 0 through positive values."},{"Start":"04:54.515 ","End":"04:57.170","Text":"You could say it\u0027s not really a limit,"},{"Start":"04:57.170 ","End":"05:00.005","Text":"but we do say that the limit is infinity, and it means something."},{"Start":"05:00.005 ","End":"05:03.150","Text":"It means this grows larger and larger."},{"Start":"05:04.280 ","End":"05:06.420","Text":"I\u0027ll leave this as the answer,"},{"Start":"05:06.420 ","End":"05:08.075","Text":"the plus is just for emphasis."},{"Start":"05:08.075 ","End":"05:09.665","Text":"You don\u0027t have to put the plus in."},{"Start":"05:09.665 ","End":"05:11.910","Text":"Anyway, we\u0027re done."}],"ID":8900},{"Watched":false,"Name":"Exercise 2 part f","Duration":"2m 42s","ChapterTopicVideoID":8545,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.760","Text":"In this exercise, we have another limit to compute with 2 variables x and y, as follows."},{"Start":"00:08.760 ","End":"00:15.675","Text":"As usual, we can\u0027t just substitute 1, 2 we\u0027ll get a 0 over 0 situation here,"},{"Start":"00:15.675 ","End":"00:20.040","Text":"because 2x plus y comes out to be 4."},{"Start":"00:20.040 ","End":"00:21.810","Text":"If we did a substitution,"},{"Start":"00:21.810 ","End":"00:28.560","Text":"we would get square root of 4 minus 3, minus 1"},{"Start":"00:28.560 ","End":"00:36.600","Text":"over 4 minus 4 and that\u0027s clearly 0 over 0."},{"Start":"00:36.600 ","End":"00:40.730","Text":"Since we don\u0027t have techniques for this in 2 variables,"},{"Start":"00:40.730 ","End":"00:43.100","Text":"we have L\u0027Hopital\u0027s rule in 1 variable,"},{"Start":"00:43.100 ","End":"00:45.020","Text":"we make a substitution."},{"Start":"00:45.020 ","End":"00:48.935","Text":"You could substitute t equals 2x plus y."},{"Start":"00:48.935 ","End":"00:50.345","Text":"That would work."},{"Start":"00:50.345 ","End":"00:53.150","Text":"I actually think we can go 1 step further"},{"Start":"00:53.150 ","End":"00:57.365","Text":"and actually do the whole 2x plus y minus 3."},{"Start":"00:57.365 ","End":"01:01.800","Text":"If we do that, it might be a little bit quicker."},{"Start":"01:02.320 ","End":"01:08.040","Text":"In this case, if x, y goes to 1,2,"},{"Start":"01:08.500 ","End":"01:12.720","Text":"I need to know what t goes to"},{"Start":"01:12.720 ","End":"01:21.585","Text":"and it goes to twice 1 is 2 plus 2 is 4, minus 3 is 1 so t goes to 1."},{"Start":"01:21.585 ","End":"01:26.130","Text":"After substituting, we get the limit of"},{"Start":"01:26.130 ","End":"01:33.765","Text":"the square root of, now this thing is just t minus 1,"},{"Start":"01:33.765 ","End":"01:38.590","Text":"and t goes to 1 over of the denominator,"},{"Start":"01:38.590 ","End":"01:43.280","Text":"I can write this minus 4 as minus 3 minus 1."},{"Start":"01:43.280 ","End":"01:47.290","Text":"We can see that this is t minus 1."},{"Start":"01:47.290 ","End":"01:52.995","Text":"Once again, we\u0027re in a 0 over 0 situation when t goes to 1."},{"Start":"01:52.995 ","End":"01:56.340","Text":"But now we have only 1 variable"},{"Start":"01:56.340 ","End":"02:02.690","Text":"and so we can use L\u0027Hopital\u0027s Rule to get an equivalent limit"},{"Start":"02:02.690 ","End":"02:10.520","Text":"or a limit that gives the same result by differentiating numerator and denominator."},{"Start":"02:10.520 ","End":"02:15.380","Text":"If we differentiate the denominator, we get 1."},{"Start":"02:15.380 ","End":"02:17.495","Text":"We differentiate the numerator,"},{"Start":"02:17.495 ","End":"02:20.960","Text":"we get 1 over twice the square root of t."},{"Start":"02:20.960 ","End":"02:22.940","Text":"That\u0027s a derivative of root t,"},{"Start":"02:22.940 ","End":"02:26.480","Text":"in case you\u0027ve forgotten, and that\u0027s it."},{"Start":"02:26.480 ","End":"02:29.960","Text":"At this point we can substitute t equals 1"},{"Start":"02:29.960 ","End":"02:34.325","Text":"and so we get 1 over twice the square root of 1."},{"Start":"02:34.325 ","End":"02:38.050","Text":"The answer is just 1/2."},{"Start":"02:38.050 ","End":"02:42.510","Text":"I\u0027ll just highlight that and declare that we\u0027re done."}],"ID":8901},{"Watched":false,"Name":"Exercise 2 part g","Duration":"3m 21s","ChapterTopicVideoID":8546,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.130","Text":"In this exercise, we have to compute a limit of expression in 2 variables, x and y."},{"Start":"00:08.130 ","End":"00:10.560","Text":"If we do the straightforward substitution,"},{"Start":"00:10.560 ","End":"00:13.770","Text":"clearly we get 1 minus 1 in the denominator,"},{"Start":"00:13.770 ","End":"00:17.430","Text":"which is 0, and on the numerator also 1 minus 1 is 0."},{"Start":"00:17.430 ","End":"00:23.820","Text":"So we have a 0 over 0 situation and so the obvious just substituting doesn\u0027t work,"},{"Start":"00:23.820 ","End":"00:27.370","Text":"so we need to use some techniques."},{"Start":"00:27.500 ","End":"00:31.230","Text":"I don\u0027t see an obvious substitution to make."},{"Start":"00:31.230 ","End":"00:37.830","Text":"What I\u0027m going to do is try and simplify the expression xy"},{"Start":"00:37.830 ","End":"00:45.450","Text":"minus y squared over the square root of x minus the square root of y and see."},{"Start":"00:45.450 ","End":"00:48.590","Text":"I\u0027m going to use the technique of the conjugate,"},{"Start":"00:48.590 ","End":"00:50.930","Text":"multiplying top and bottom by the conjugate."},{"Start":"00:50.930 ","End":"00:57.380","Text":"What we do is we multiply square root of x plus square root of y,"},{"Start":"00:57.380 ","End":"01:04.855","Text":"because when we have something like A plus B,"},{"Start":"01:04.855 ","End":"01:09.405","Text":"then its conjugate will be A minus B and vice versa,"},{"Start":"01:09.405 ","End":"01:11.300","Text":"A minus B, its conjugate."},{"Start":"01:11.300 ","End":"01:13.880","Text":"If you multiply something by its conjugate,"},{"Start":"01:13.880 ","End":"01:18.960","Text":"you get the difference of squares formula A squared minus B squared."},{"Start":"01:18.960 ","End":"01:21.050","Text":"Let\u0027s see what happens here."},{"Start":"01:21.050 ","End":"01:24.680","Text":"Square root of x plus square root of y that,"},{"Start":"01:24.680 ","End":"01:28.960","Text":"that equals, let me just move this out of the way."},{"Start":"01:28.960 ","End":"01:34.220","Text":"The denominator is the interesting bit because here we do have well,"},{"Start":"01:34.220 ","End":"01:36.545","Text":"it\u0027s in the reverse order, A minus B, A plus B."},{"Start":"01:36.545 ","End":"01:45.050","Text":"So we get the square root of x squared minus the square root of y squared."},{"Start":"01:45.050 ","End":"01:48.980","Text":"On the top, here I can take y outside the brackets,"},{"Start":"01:48.980 ","End":"01:53.585","Text":"that might help y times x minus y"},{"Start":"01:53.585 ","End":"01:59.810","Text":"and here I have square root of x plus square root of y."},{"Start":"01:59.810 ","End":"02:02.345","Text":"Let\u0027s see what we get."},{"Start":"02:02.345 ","End":"02:09.420","Text":"This is going to equal, well,"},{"Start":"02:09.420 ","End":"02:14.620","Text":"the numerator as is but,"},{"Start":"02:15.040 ","End":"02:18.590","Text":"I just copy pasted it to show you what I\u0027m doing,"},{"Start":"02:18.590 ","End":"02:24.620","Text":"the denominator, where I see the square root of x squared."},{"Start":"02:24.620 ","End":"02:29.735","Text":"I can put x and where I\u0027ve got the square root of y squared, I can put y."},{"Start":"02:29.735 ","End":"02:36.150","Text":"Now look, this term cancels with the denominator."},{"Start":"02:36.150 ","End":"02:39.180","Text":"Now that I\u0027ve done the simplification of this expression,"},{"Start":"02:39.180 ","End":"02:40.430","Text":"I\u0027m going back to the limit."},{"Start":"02:40.430 ","End":"02:45.110","Text":"I\u0027m going to the limit as x,"},{"Start":"02:45.110 ","End":"02:47.630","Text":"y goes to 1,"},{"Start":"02:47.630 ","End":"02:58.855","Text":"1 of y times square root of x plus square root of y."},{"Start":"02:58.855 ","End":"03:02.330","Text":"At this stage, there\u0027s no problem in substituting."},{"Start":"03:02.330 ","End":"03:05.175","Text":"So I just get, let\u0027s see,"},{"Start":"03:05.175 ","End":"03:10.925","Text":"1 times square root of 1 plus square root of 1."},{"Start":"03:10.925 ","End":"03:22.200","Text":"That\u0027s 1 times 1 plus 1 and so the answer is 2 and we are done."}],"ID":8902},{"Watched":false,"Name":"Exercise 2 part h","Duration":"4m 24s","ChapterTopicVideoID":8547,"CourseChapterTopicPlaylistID":4970,"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":"This time we have a limit in 3 variables,"},{"Start":"00:03.015 ","End":"00:05.530","Text":"x, y and z."},{"Start":"00:06.020 ","End":"00:11.940","Text":"As usual, you can\u0027t expect it to be so easy as just a substitute x,"},{"Start":"00:11.940 ","End":"00:12.990","Text":"y, and z. I mean,"},{"Start":"00:12.990 ","End":"00:18.530","Text":"x goes to 0 so a numerator goes to 0 and sine 0 is"},{"Start":"00:18.530 ","End":"00:20.660","Text":"0 and the denominator 0 so we have"},{"Start":"00:20.660 ","End":"00:26.110","Text":"a 0 over 0 situation so it\u0027s going to be a little trickier."},{"Start":"00:26.110 ","End":"00:29.400","Text":"In fact, the usual tricks don\u0027t seem to work here."},{"Start":"00:29.400 ","End":"00:32.130","Text":"A straightforward substitution doesn\u0027t work,"},{"Start":"00:32.130 ","End":"00:34.755","Text":"so we\u0027re going to use an extra trick."},{"Start":"00:34.755 ","End":"00:36.840","Text":"Let me first write it out."},{"Start":"00:36.840 ","End":"00:44.280","Text":"We have the limit as x, y, z goes to,"},{"Start":"00:44.280 ","End":"00:47.310","Text":"I will write it underneath, 0, 1,"},{"Start":"00:47.310 ","End":"00:53.790","Text":"2 of sine x times y"},{"Start":"00:53.790 ","End":"01:01.935","Text":"squared plus z squared over xy squared."},{"Start":"01:01.935 ","End":"01:06.690","Text":"Now, if on the denominator I also had x times"},{"Start":"01:06.690 ","End":"01:11.565","Text":"y squared plus z squared then I\u0027d know how to substitute."},{"Start":"01:11.565 ","End":"01:17.075","Text":"Here\u0027s the trick. Let\u0027s put here what we want to have,"},{"Start":"01:17.075 ","End":"01:22.025","Text":"which is y squared plus z squared."},{"Start":"01:22.025 ","End":"01:24.290","Text":"But obviously I can\u0027t just go around changing things,"},{"Start":"01:24.290 ","End":"01:26.030","Text":"so what I\u0027m going to do is fix it up."},{"Start":"01:26.030 ","End":"01:33.469","Text":"If I put x times y squared plus z squared on the numerator of a fraction,"},{"Start":"01:33.469 ","End":"01:39.225","Text":"it will cancel out and I will put back here the original xy squared."},{"Start":"01:39.225 ","End":"01:42.905","Text":"Now this cancel with this and we\u0027re left with the original."},{"Start":"01:42.905 ","End":"01:46.220","Text":"What we do now is we separate it into 2 limits."},{"Start":"01:46.220 ","End":"01:50.095","Text":"We have the limit of this bit."},{"Start":"01:50.095 ","End":"01:52.455","Text":"Let me write this out."},{"Start":"01:52.455 ","End":"01:55.260","Text":"The first 1 is the limit,"},{"Start":"01:55.260 ","End":"01:58.560","Text":"let\u0027s see if I can squeeze it in, x, y,"},{"Start":"01:58.560 ","End":"02:02.730","Text":"z goes to 0, 1, 2."},{"Start":"02:02.730 ","End":"02:04.875","Text":"I\u0027ll just copy this bit."},{"Start":"02:04.875 ","End":"02:09.385","Text":"Then multiplied by the limit of this thing."},{"Start":"02:09.385 ","End":"02:15.875","Text":"Here, actually just copy this bit here and this bit here and save myself writing."},{"Start":"02:15.875 ","End":"02:18.485","Text":"Let\u0027s take each 1 separately."},{"Start":"02:18.485 ","End":"02:22.200","Text":"The first 1, this 1 here,"},{"Start":"02:22.200 ","End":"02:31.055","Text":"we use our usual trick of letting t equal x times y squared plus z squared."},{"Start":"02:31.055 ","End":"02:32.810","Text":"Then when x, y,"},{"Start":"02:32.810 ","End":"02:34.400","Text":"z goes to 0, 1,"},{"Start":"02:34.400 ","End":"02:40.170","Text":"2, x goes to 0 so t goes to"},{"Start":"02:40.170 ","End":"02:47.730","Text":"0 so we get the limit as t goes to 0,"},{"Start":"02:47.730 ","End":"02:57.395","Text":"sine of t over t. This limit we\u0027ve done many times before so I\u0027ll just quote the result,"},{"Start":"02:57.395 ","End":"02:59.900","Text":"It\u0027s equal to 1."},{"Start":"02:59.900 ","End":"03:03.620","Text":"If you\u0027re not sure, you can always go into L\u0027Hospital\u0027s rule,"},{"Start":"03:03.620 ","End":"03:08.870","Text":"cosine t over 1 substitute t equals 0, that\u0027s what we get."},{"Start":"03:08.870 ","End":"03:16.730","Text":"As for the other 1, well,"},{"Start":"03:16.730 ","End":"03:19.129","Text":"it turns out what I did was partially redundant,"},{"Start":"03:19.129 ","End":"03:21.560","Text":"not wrong but I didn\u0027t need the x."},{"Start":"03:21.560 ","End":"03:24.035","Text":"In fact, I can also cancel it out here."},{"Start":"03:24.035 ","End":"03:27.400","Text":"So away with that and away with that."},{"Start":"03:27.400 ","End":"03:30.050","Text":"I should have done this in the first place."},{"Start":"03:30.050 ","End":"03:38.190","Text":"In fact, I can just strike it out here and here and it still just works fine."},{"Start":"03:39.140 ","End":"03:42.530","Text":"At this point, there\u0027s no problem."},{"Start":"03:42.530 ","End":"03:45.305","Text":"I can just substitute x, y, and z."},{"Start":"03:45.305 ","End":"03:51.690","Text":"Well, there is no x to substitute but if I let y equals"},{"Start":"03:51.690 ","End":"03:58.744","Text":"1 and z equals 2 in this expression then x equals 0 but there is no way to substitute."},{"Start":"03:58.744 ","End":"04:07.390","Text":"Then what I get is 1 squared plus 2 squared over 1 squared."},{"Start":"04:07.390 ","End":"04:11.145","Text":"That is if 1 plus 4 over 1, that is 5."},{"Start":"04:11.145 ","End":"04:17.070","Text":"My answer is 1 from here and 5 from here"},{"Start":"04:17.070 ","End":"04:22.110","Text":"multiplied, so altogether I get 5."},{"Start":"04:22.110 ","End":"04:25.330","Text":"That\u0027s the answer and we\u0027re done."}],"ID":8903},{"Watched":false,"Name":"Exercise 3 part a","Duration":"3m 44s","ChapterTopicVideoID":8554,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.740","Text":"In this exercise, we have to compute or at least try to compute the following limit."},{"Start":"00:07.740 ","End":"00:10.455","Text":"Let\u0027s see why there\u0027s even a problem."},{"Start":"00:10.455 ","End":"00:16.140","Text":"The obvious thing that we normally do is to substitute x equals 0,"},{"Start":"00:16.140 ","End":"00:17.520","Text":"y equals 0 here,"},{"Start":"00:17.520 ","End":"00:18.600","Text":"but then if we do that,"},{"Start":"00:18.600 ","End":"00:20.360","Text":"we get 0 over 0,"},{"Start":"00:20.360 ","End":"00:22.990","Text":"we know there\u0027s a problem here."},{"Start":"00:22.990 ","End":"00:28.860","Text":"There\u0027s no obvious technique like substitution that will work here,"},{"Start":"00:28.860 ","End":"00:32.685","Text":"in fact, it turns out that this limit doesn\u0027t exist."},{"Start":"00:32.685 ","End":"00:37.740","Text":"One of the techniques we have for showing the limit doesn\u0027t exist."},{"Start":"00:37.740 ","End":"00:40.260","Text":"I think it was technique number 3 in any way,"},{"Start":"00:40.260 ","End":"00:44.969","Text":"any event it was called the iterated limit technique,"},{"Start":"00:44.969 ","End":"00:48.175","Text":"and it\u0027s used for non-existence of a limit."},{"Start":"00:48.175 ","End":"00:50.630","Text":"The iterated limit means instead of doing"},{"Start":"00:50.630 ","End":"00:54.605","Text":"a double limit where x and y simultaneously go to 0,"},{"Start":"00:54.605 ","End":"00:56.765","Text":"we take them one at a time,"},{"Start":"00:56.765 ","End":"00:59.275","Text":"first x and then y,"},{"Start":"00:59.275 ","End":"01:01.115","Text":"and we compute the limit,"},{"Start":"01:01.115 ","End":"01:02.360","Text":"and then we do the opposite."},{"Start":"01:02.360 ","End":"01:03.950","Text":"First y and then x,"},{"Start":"01:03.950 ","End":"01:05.750","Text":"and if we get 2 different answers,"},{"Start":"01:05.750 ","End":"01:08.390","Text":"then we know that the limit doesn\u0027t exist."},{"Start":"01:08.390 ","End":"01:12.230","Text":"In this case, what we\u0027re doing is we take the limit,"},{"Start":"01:12.230 ","End":"01:13.910","Text":"doesn\u0027t matter which we do first."},{"Start":"01:13.910 ","End":"01:16.265","Text":"Let\u0027s say in this one,"},{"Start":"01:16.265 ","End":"01:20.210","Text":"the one we do second is the outer one,"},{"Start":"01:20.210 ","End":"01:22.745","Text":"and then the inner one,"},{"Start":"01:22.745 ","End":"01:24.470","Text":"x goes to 0,"},{"Start":"01:24.470 ","End":"01:25.970","Text":"and I\u0027ll put a bracket here."},{"Start":"01:25.970 ","End":"01:36.380","Text":"We let x go to 0 for x squared plus y squared squared over x_fourth plus y squared."},{"Start":"01:36.380 ","End":"01:38.930","Text":"We let x go to 0, we get a function of y,"},{"Start":"01:38.930 ","End":"01:40.370","Text":"and then we let y go to 0."},{"Start":"01:40.370 ","End":"01:42.890","Text":"The question is, does this equal?"},{"Start":"01:42.890 ","End":"01:45.170","Text":"I claim that not,"},{"Start":"01:45.170 ","End":"01:50.600","Text":"the other way around where the second limit is letting x go to 0,"},{"Start":"01:50.600 ","End":"01:54.640","Text":"and the first limit is y goes to 0,"},{"Start":"01:54.640 ","End":"01:59.300","Text":"and then I just copied this expression from here to here."},{"Start":"01:59.300 ","End":"02:02.330","Text":"Let\u0027s start with the left one."},{"Start":"02:02.330 ","End":"02:04.685","Text":"We let x go to 0,"},{"Start":"02:04.685 ","End":"02:07.640","Text":"we can just substitute x equals 0 here,"},{"Start":"02:07.640 ","End":"02:12.680","Text":"and then we get y squared squared,"},{"Start":"02:12.680 ","End":"02:17.925","Text":"which is y_fourth on the numerator,"},{"Start":"02:17.925 ","End":"02:21.140","Text":"and on the denominator, if x goes to 0,"},{"Start":"02:21.140 ","End":"02:28.300","Text":"we get y squared limit as y goes to 0,"},{"Start":"02:28.300 ","End":"02:38.415","Text":"and this is equal to the limit as y goes to 0 of y squared,"},{"Start":"02:38.415 ","End":"02:42.045","Text":"and this is just 0."},{"Start":"02:42.045 ","End":"02:45.390","Text":"This is what happens when we develop the left-hand side."},{"Start":"02:45.390 ","End":"02:48.635","Text":"Now, let\u0027s see what happens on the right-hand side."},{"Start":"02:48.635 ","End":"02:54.330","Text":"This is equal to the limit as x goes to 0,"},{"Start":"02:54.330 ","End":"02:57.405","Text":"and now here I let y go to 0."},{"Start":"02:57.405 ","End":"03:00.410","Text":"What we get is x squared squared,"},{"Start":"03:00.410 ","End":"03:03.290","Text":"which is x_fourth, on the bottom,"},{"Start":"03:03.290 ","End":"03:06.430","Text":"y is 0, it\u0027s also x_fourth."},{"Start":"03:06.430 ","End":"03:11.325","Text":"This is the limit as x goes to 0,"},{"Start":"03:11.325 ","End":"03:13.370","Text":"this over this is 1,"},{"Start":"03:13.370 ","End":"03:16.380","Text":"and the answer is 1."},{"Start":"03:16.450 ","End":"03:20.580","Text":"Clearly, this does not equal 0,"},{"Start":"03:20.580 ","End":"03:22.620","Text":"does not equal to 1,"},{"Start":"03:22.620 ","End":"03:27.020","Text":"because the iterated limits are not equal,"},{"Start":"03:27.020 ","End":"03:29.419","Text":"the limit does not exist."},{"Start":"03:29.419 ","End":"03:33.145","Text":"We can just say that the answer is,"},{"Start":"03:33.145 ","End":"03:44.940","Text":"does not exist and leave it at that. We\u0027re done."}],"ID":8904},{"Watched":false,"Name":"Exercise 3 part b","Duration":"3m 5s","ChapterTopicVideoID":8555,"CourseChapterTopicPlaylistID":4970,"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\u0027re going to try and compute the following limit,"},{"Start":"00:04.500 ","End":"00:08.024","Text":"x and y both go to 0,"},{"Start":"00:08.024 ","End":"00:12.960","Text":"absolute value of y^x."},{"Start":"00:12.960 ","End":"00:20.100","Text":"We can\u0027t just substitute because 0^0 is 1 of those indeterminate forms,"},{"Start":"00:20.100 ","End":"00:24.360","Text":"undefined, and there\u0027s no obvious technique."},{"Start":"00:24.360 ","End":"00:27.390","Text":"It turns out that this limit actually doesn\u0027t exist,"},{"Start":"00:27.390 ","End":"00:34.530","Text":"and remember we have a technique for non-existence called the iterated limit technique,"},{"Start":"00:34.530 ","End":"00:38.530","Text":"so I\u0027ll just write that down, iterated limit."},{"Start":"00:38.530 ","End":"00:44.930","Text":"This technique involves comparing 2 iterated limits."},{"Start":"00:44.930 ","End":"00:50.424","Text":"Iterated means first letting x go to 0 and then letting y go to 0,"},{"Start":"00:50.424 ","End":"00:51.900","Text":"or the other way round,"},{"Start":"00:51.900 ","End":"00:56.310","Text":"there\u0027s 2 iterated limits first x and y or first y then x."},{"Start":"00:56.310 ","End":"00:59.810","Text":"If those 2 don\u0027t agree,"},{"Start":"00:59.810 ","End":"01:02.450","Text":"they\u0027re different, then there\u0027s no limit."},{"Start":"01:02.450 ","End":"01:07.495","Text":"What we have to show or at least compare is the limit."},{"Start":"01:07.495 ","End":"01:10.425","Text":"It doesn\u0027t matter which way round,"},{"Start":"01:10.425 ","End":"01:12.575","Text":"here limit x goes to 0,"},{"Start":"01:12.575 ","End":"01:19.890","Text":"limit y goes to 0 of absolute value of y^x."},{"Start":"01:20.690 ","End":"01:23.385","Text":"I don\u0027t know if this equals or not,"},{"Start":"01:23.385 ","End":"01:26.110","Text":"I\u0027m claiming that it doesn\u0027t equal,"},{"Start":"01:26.110 ","End":"01:35.370","Text":"the limit y goes to 0 of the limit as x goes to 0 of the same thing."},{"Start":"01:35.590 ","End":"01:38.255","Text":"Let\u0027s start with the left-hand side."},{"Start":"01:38.255 ","End":"01:40.370","Text":"The inner 1 is what we compute first."},{"Start":"01:40.370 ","End":"01:43.310","Text":"We first let y go to 0,"},{"Start":"01:43.310 ","End":"01:47.210","Text":"and so we\u0027ve got the limit as x goes to 0."},{"Start":"01:47.210 ","End":"01:49.680","Text":"Then if y goes to 0,"},{"Start":"01:50.200 ","End":"01:56.310","Text":"then we get 0^x and x is not 0,"},{"Start":"01:56.310 ","End":"01:57.990","Text":"it\u0027s only approaching 0,"},{"Start":"01:57.990 ","End":"02:02.375","Text":"and 0 to the power of anything is 0."},{"Start":"02:02.375 ","End":"02:05.180","Text":"I\u0027ll just make a little note that we have 0^x."},{"Start":"02:05.180 ","End":"02:10.045","Text":"So the limit of x goes to 0 of 0,"},{"Start":"02:10.045 ","End":"02:13.710","Text":"and the limit of x goes to 0, well,"},{"Start":"02:13.710 ","End":"02:17.615","Text":"there\u0027s no x here, of a constant 0 is just 0."},{"Start":"02:17.615 ","End":"02:19.640","Text":"This is developing the left-hand side,"},{"Start":"02:19.640 ","End":"02:21.980","Text":"now let\u0027s develop the right-hand side."},{"Start":"02:21.980 ","End":"02:26.810","Text":"We\u0027ve got the limit as y goes to 0."},{"Start":"02:26.810 ","End":"02:29.240","Text":"Now if x goes to 0,"},{"Start":"02:29.240 ","End":"02:35.755","Text":"we have absolute value of y^0 here if we substitute,"},{"Start":"02:35.755 ","End":"02:39.490","Text":"and they anything to the power of 0 is 1."},{"Start":"02:39.490 ","End":"02:42.405","Text":"The limit of y goes to 0 of 1,"},{"Start":"02:42.405 ","End":"02:44.750","Text":"again it\u0027s a constant as far as y goes,"},{"Start":"02:44.750 ","End":"02:46.550","Text":"so the answer is 1."},{"Start":"02:46.550 ","End":"02:48.875","Text":"If we compare these 2,"},{"Start":"02:48.875 ","End":"02:51.860","Text":"they are definitely not equal."},{"Start":"02:51.860 ","End":"02:53.765","Text":"Because they are not equal,"},{"Start":"02:53.765 ","End":"02:58.310","Text":"we can say that the original limit does"},{"Start":"02:58.310 ","End":"03:05.470","Text":"not exist and we just leave the answer as that. We\u0027re done."}],"ID":8905},{"Watched":false,"Name":"Exercise 3 part c","Duration":"3m 5s","ChapterTopicVideoID":8556,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"In this exercise, we have to compute a limit of a function of 2 variables,"},{"Start":"00:04.860 ","End":"00:07.305","Text":"as x and y both go to 0."},{"Start":"00:07.305 ","End":"00:10.155","Text":"The obvious thing to do is to substitute,"},{"Start":"00:10.155 ","End":"00:13.680","Text":"but that doesn\u0027t work because we get 0 over 0."},{"Start":"00:13.680 ","End":"00:17.310","Text":"We looked through our bag of tricks to try and find the limit. Nothing helped."},{"Start":"00:17.310 ","End":"00:20.740","Text":"Turns out this has no limit."},{"Start":"00:20.740 ","End":"00:25.260","Text":"The main technique we have for disproving"},{"Start":"00:25.260 ","End":"00:30.060","Text":"that there\u0027s a limit is the iterative limit method."},{"Start":"00:30.060 ","End":"00:34.590","Text":"I\u0027ll just write that word iterative."},{"Start":"00:34.590 ","End":"00:36.640","Text":"There are actually 2 iterative limits."},{"Start":"00:36.640 ","End":"00:41.655","Text":"One, we first let x go to 0 and then y go to 0, in this case."},{"Start":"00:41.655 ","End":"00:43.340","Text":"Then the other iterative limit,"},{"Start":"00:43.340 ","End":"00:46.965","Text":"we first go for y and then x."},{"Start":"00:46.965 ","End":"00:50.135","Text":"If these 2 iterative limits are not equal,"},{"Start":"00:50.135 ","End":"00:53.045","Text":"then this thing has no limit. Let\u0027s see."},{"Start":"00:53.045 ","End":"00:55.624","Text":"Let\u0027s check if the limit,"},{"Start":"00:55.624 ","End":"00:57.140","Text":"the 2 iterative limits,"},{"Start":"00:57.140 ","End":"01:01.250","Text":"one of them is x goes to 0 here,"},{"Start":"01:01.250 ","End":"01:04.865","Text":"and then y goes to 0 here."},{"Start":"01:04.865 ","End":"01:11.420","Text":"Same thing, x cubed plus y squared over x squared plus y squared."},{"Start":"01:11.420 ","End":"01:17.000","Text":"But it\u0027s in this 1 to y is taken first and then the x we work from inside out."},{"Start":"01:17.000 ","End":"01:19.165","Text":"Now, does this equal?"},{"Start":"01:19.165 ","End":"01:20.760","Text":"I claim not."},{"Start":"01:20.760 ","End":"01:23.180","Text":"To the other way around, first,"},{"Start":"01:23.180 ","End":"01:25.385","Text":"we take y goes to 0,"},{"Start":"01:25.385 ","End":"01:28.850","Text":"and then x goes to 0."},{"Start":"01:28.850 ","End":"01:30.695","Text":"Same thing as here,"},{"Start":"01:30.695 ","End":"01:32.240","Text":"which I just copied."},{"Start":"01:32.240 ","End":"01:35.180","Text":"Now, let\u0027s work on the left-hand side."},{"Start":"01:35.180 ","End":"01:42.470","Text":"This thing is equal to the limit as x goes to 0."},{"Start":"01:42.470 ","End":"01:46.255","Text":"Notice that when x goes to 0, x isn\u0027t 0."},{"Start":"01:46.255 ","End":"01:51.500","Text":"If x is not 0, we can compute this limit by substituting y equals 0."},{"Start":"01:51.500 ","End":"01:54.380","Text":"So we get x cubed over x squared."},{"Start":"01:54.380 ","End":"01:58.320","Text":"It\u0027s not 0 over 0, limit exists."},{"Start":"01:58.320 ","End":"02:01.500","Text":"x is close to 0 but not 0."},{"Start":"02:01.500 ","End":"02:03.285","Text":"Let\u0027s continue with this."},{"Start":"02:03.285 ","End":"02:12.260","Text":"This is the limit as x goes to 0 of just x and that is just 0."},{"Start":"02:12.260 ","End":"02:15.605","Text":"Let\u0027s see what we get on the right-hand side."},{"Start":"02:15.605 ","End":"02:18.410","Text":"y goes to 0, so y is not 0."},{"Start":"02:18.410 ","End":"02:22.430","Text":"If y is not 0, we can just substitute x equals 0 here."},{"Start":"02:22.430 ","End":"02:29.480","Text":"This limit becomes y squared over y squared."},{"Start":"02:29.480 ","End":"02:33.965","Text":"The limit as y goes to 0."},{"Start":"02:33.965 ","End":"02:37.630","Text":"Now, when y is not 0, just tends to 0,"},{"Start":"02:37.630 ","End":"02:39.070","Text":"this thing is equal to 1,"},{"Start":"02:39.070 ","End":"02:43.315","Text":"so we get the limit as y goes to 0 of 1."},{"Start":"02:43.315 ","End":"02:46.070","Text":"The limit of a constant is that constant,"},{"Start":"02:46.070 ","End":"02:47.480","Text":"so we get 1."},{"Start":"02:47.480 ","End":"02:51.650","Text":"These 2 are definitely not equal."},{"Start":"02:51.650 ","End":"02:54.650","Text":"Because these 2 iterative limits are not equal,"},{"Start":"02:54.650 ","End":"03:05.370","Text":"this thing has no limit or the limit does not exist or whatever. We\u0027re done."}],"ID":8906},{"Watched":false,"Name":"Exercise 3 part d","Duration":"5m 42s","ChapterTopicVideoID":8557,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.319","Text":"In this exercise, we have to compute the following limit function of 2 variables,"},{"Start":"00:04.319 ","End":"00:08.325","Text":"x over y when x and y go both to 0."},{"Start":"00:08.325 ","End":"00:15.375","Text":"Now, we can\u0027t just substitute x and y equals 0 because then we\u0027ll get 0 over 0."},{"Start":"00:15.375 ","End":"00:18.670","Text":"If you think about it awhile, nothing really seems to work,"},{"Start":"00:18.670 ","End":"00:21.495","Text":"so we suspect that maybe there is no limit."},{"Start":"00:21.495 ","End":"00:24.330","Text":"The first thing it tried to do to prove no limit,"},{"Start":"00:24.330 ","End":"00:27.975","Text":"the easier 1 is the iterated limit technique."},{"Start":"00:27.975 ","End":"00:33.895","Text":"That\u0027s not going to work either because there is no iterated limit in 1 direction."},{"Start":"00:33.895 ","End":"00:36.925","Text":"For example, if I take the limit,"},{"Start":"00:36.925 ","End":"00:39.880","Text":"or I try to take the limit."},{"Start":"00:40.610 ","End":"00:44.480","Text":"First, meaning I write it first but it\u0027s the second,"},{"Start":"00:44.480 ","End":"00:46.985","Text":"it\u0027s outward of limit,"},{"Start":"00:46.985 ","End":"00:52.325","Text":"y goes to 0 of x over y."},{"Start":"00:52.325 ","End":"00:56.420","Text":"We\u0027re already stuck because this thing has no limit."},{"Start":"00:56.420 ","End":"01:00.360","Text":"If x is not 0, denominator goes to 0,"},{"Start":"01:00.360 ","End":"01:02.990","Text":"it has no limit, it\u0027s not even infinity,"},{"Start":"01:02.990 ","End":"01:05.795","Text":"you can\u0027t write plus or minus infinity."},{"Start":"01:05.795 ","End":"01:09.680","Text":"The limit of something non-zero over something tends to 0 has no limit,"},{"Start":"01:09.680 ","End":"01:13.015","Text":"so this thing, basically,"},{"Start":"01:13.015 ","End":"01:16.515","Text":"it throughout our technique."},{"Start":"01:16.515 ","End":"01:20.290","Text":"We\u0027re going to have to try more powerful tool,"},{"Start":"01:20.290 ","End":"01:25.160","Text":"and the more powerful tool I called it number 3, the iterative limits,"},{"Start":"01:25.160 ","End":"01:27.380","Text":"and there\u0027s also another 1,"},{"Start":"01:27.380 ","End":"01:33.900","Text":"I called it the limit along a path,"},{"Start":"01:33.900 ","End":"01:36.669","Text":"or really 2 different paths."},{"Start":"01:36.669 ","End":"01:40.790","Text":"The idea is that we take the point,"},{"Start":"01:40.790 ","End":"01:44.885","Text":"in this case 0,0 and we take 2 different paths to the point,"},{"Start":"01:44.885 ","End":"01:47.680","Text":"and we take the limit along the path."},{"Start":"01:47.680 ","End":"01:51.650","Text":"If we can find 2 paths which give different limits,"},{"Start":"01:51.650 ","End":"01:56.375","Text":"then we can say that this limit doesn\u0027t exist."},{"Start":"01:56.375 ","End":"02:00.439","Text":"Let me illustrate with a picture in case you\u0027ve forgotten the technique."},{"Start":"02:00.439 ","End":"02:04.295","Text":"The point we have is the origin 0,0."},{"Start":"02:04.295 ","End":"02:06.635","Text":"We\u0027ll take 2 different paths,"},{"Start":"02:06.635 ","End":"02:09.294","Text":"could be straight lines,"},{"Start":"02:09.294 ","End":"02:17.195","Text":"1 line might be here and we\u0027d go along this path to the 0,0."},{"Start":"02:17.195 ","End":"02:22.024","Text":"Another 1 might be to go from here,"},{"Start":"02:22.024 ","End":"02:29.510","Text":"and also in this direction until we get to 0,0."},{"Start":"02:29.510 ","End":"02:32.485","Text":"Then we check if the 2 limits are equal."},{"Start":"02:32.485 ","End":"02:35.375","Text":"If we choose properly,"},{"Start":"02:35.375 ","End":"02:41.030","Text":"we should be able to find 2 for which the limits are different."},{"Start":"02:41.030 ","End":"02:44.510","Text":"Usually the path is given as y as a function of x,"},{"Start":"02:44.510 ","End":"02:47.350","Text":"though it could be parametrized or x as a function of y,"},{"Start":"02:47.350 ","End":"02:49.550","Text":"but almost always you can settle for,"},{"Start":"02:49.550 ","End":"02:52.220","Text":"let\u0027s say this is y equals 2x,"},{"Start":"02:52.220 ","End":"02:59.860","Text":"I\u0027m just guessing, and let\u0027s say that this 1 is y equals x."},{"Start":"03:00.850 ","End":"03:05.030","Text":"Notice that whichever path we choose,"},{"Start":"03:05.030 ","End":"03:07.370","Text":"when x goes to 0,"},{"Start":"03:07.370 ","End":"03:11.490","Text":"then y also goes to 0,"},{"Start":"03:11.490 ","End":"03:14.390","Text":"whether it\u0027s x or 2x, it goes to 0."},{"Start":"03:14.390 ","End":"03:16.420","Text":"In general, this doesn\u0027t have to be 0,0,"},{"Start":"03:16.420 ","End":"03:18.960","Text":"if we had another point A,"},{"Start":"03:18.960 ","End":"03:20.880","Text":"B, when x goes to A,"},{"Start":"03:20.880 ","End":"03:25.220","Text":"y has to go to B, has to choose the path that it goes to that point."},{"Start":"03:25.220 ","End":"03:32.570","Text":"Let\u0027s use these 2 examples and see what the limit is along each of these paths."},{"Start":"03:32.570 ","End":"03:35.345","Text":"Let\u0027s try the green 1 first."},{"Start":"03:35.345 ","End":"03:44.305","Text":"We want the limit along y equals 2x."},{"Start":"03:44.305 ","End":"03:47.795","Text":"I\u0027ll just emphasize that x goes to 0."},{"Start":"03:47.795 ","End":"03:56.535","Text":"What we get is the limit as x goes to 0,"},{"Start":"03:56.535 ","End":"04:02.080","Text":"and I\u0027ll just remind myself or even write it in here that y equals to 2x,"},{"Start":"04:02.080 ","End":"04:08.810","Text":"that\u0027s the path that I\u0027m choosing of x over y is equal to."},{"Start":"04:08.810 ","End":"04:13.160","Text":"We just take the limit as x goes to 0,"},{"Start":"04:13.160 ","End":"04:15.725","Text":"instead of y, we substitute 2x,"},{"Start":"04:15.725 ","End":"04:17.810","Text":"so it\u0027s x over 2x."},{"Start":"04:17.810 ","End":"04:26.145","Text":"X over 2x,"},{"Start":"04:26.145 ","End":"04:30.570","Text":"the x cancels and I\u0027m just left with 1 in the numerator."},{"Start":"04:30.570 ","End":"04:33.415","Text":"We get the limit of a constant 1/2,"},{"Start":"04:33.415 ","End":"04:35.800","Text":"and this is just 1/2."},{"Start":"04:35.800 ","End":"04:38.740","Text":"Along this 1, we got the limit of 1/2."},{"Start":"04:38.740 ","End":"04:40.555","Text":"Let\u0027s try the other path."},{"Start":"04:40.555 ","End":"04:48.295","Text":"Here we have the limit as x goes to 0,"},{"Start":"04:48.295 ","End":"04:51.744","Text":"but along y equals x,"},{"Start":"04:51.744 ","End":"04:54.370","Text":"also of y over x."},{"Start":"04:54.370 ","End":"04:56.635","Text":"This time it\u0027s equal to the limit,"},{"Start":"04:56.635 ","End":"04:59.470","Text":"as x goes to 0,"},{"Start":"04:59.470 ","End":"05:01.240","Text":"y is equal to x,"},{"Start":"05:01.240 ","End":"05:04.220","Text":"so we have x over x,"},{"Start":"05:04.220 ","End":"05:09.150","Text":"and x over x, it cancels out,"},{"Start":"05:09.150 ","End":"05:11.190","Text":"it\u0027s just equal to 1,"},{"Start":"05:11.190 ","End":"05:15.250","Text":"and so this limit is equal to 1."},{"Start":"05:15.670 ","End":"05:21.995","Text":"Along 1 path we got a 1/2 and along the other path we got 1."},{"Start":"05:21.995 ","End":"05:25.345","Text":"Since these 2 are not equal,"},{"Start":"05:25.345 ","End":"05:32.290","Text":"then we can say that this limit does not exist,"},{"Start":"05:32.290 ","End":"05:38.255","Text":"or there is no limit or something like that does not exist."},{"Start":"05:38.255 ","End":"05:42.300","Text":"Having said that, we are done."}],"ID":8907},{"Watched":false,"Name":"Exercise 3 part e","Duration":"6m 9s","ChapterTopicVideoID":8558,"CourseChapterTopicPlaylistID":4970,"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 compute the limit of 2 variables as x,"},{"Start":"00:04.020 ","End":"00:07.180","Text":"y go to 0, 0."},{"Start":"00:07.550 ","End":"00:11.040","Text":"The obvious thing to do is to try and substitute first."},{"Start":"00:11.040 ","End":"00:12.360","Text":"That doesn\u0027t give us anything."},{"Start":"00:12.360 ","End":"00:14.175","Text":"It gives us 0 over 0."},{"Start":"00:14.175 ","End":"00:19.485","Text":"In fact, I even tried the iterative limits,"},{"Start":"00:19.485 ","End":"00:21.675","Text":"neither of them exists."},{"Start":"00:21.675 ","End":"00:34.290","Text":"So our hope is the limit"},{"Start":"00:34.290 ","End":"00:35.535","Text":"along a path."},{"Start":"00:35.535 ","End":"00:38.250","Text":"We tried to find 2 different paths to 0,"},{"Start":"00:38.250 ","End":"00:42.345","Text":"0, which give 2 different limits."},{"Start":"00:42.345 ","End":"00:44.640","Text":"In the previous exercise,"},{"Start":"00:44.640 ","End":"00:46.250","Text":"we tried 2 paths,"},{"Start":"00:46.250 ","End":"00:49.850","Text":"straight lines, y equals x and y equals 2x."},{"Start":"00:49.850 ","End":"00:51.020","Text":"It was good for us then,"},{"Start":"00:51.020 ","End":"00:52.490","Text":"maybe it will work here."},{"Start":"00:52.490 ","End":"00:58.610","Text":"Let\u0027s try. Let\u0027s take the red path first, y equals x."},{"Start":"00:58.610 ","End":"01:08.015","Text":"We get the limit as x goes to 0 and y equals x,"},{"Start":"01:08.015 ","End":"01:09.620","Text":"which of course also goes to 0."},{"Start":"01:09.620 ","End":"01:16.520","Text":"Therefore, of x squared y over x to the fourth plus y squared."},{"Start":"01:16.520 ","End":"01:21.980","Text":"What it equals as we put y equals x and we get the limit of a single variable,"},{"Start":"01:21.980 ","End":"01:26.670","Text":"x, x squared times y is x cubed."},{"Start":"01:27.200 ","End":"01:33.830","Text":"x to the fourth plus y squared is x to the fourth plus x squared."},{"Start":"01:33.830 ","End":"01:36.904","Text":"x just goes to 0."},{"Start":"01:36.904 ","End":"01:42.935","Text":"It\u0027s not equal to 0, the best we can do is x squared."},{"Start":"01:42.935 ","End":"01:46.355","Text":"So if I divide top and bottom by x squared,"},{"Start":"01:46.355 ","End":"01:49.355","Text":"this will be left with just x."},{"Start":"01:49.355 ","End":"01:51.530","Text":"This will be 1,"},{"Start":"01:51.530 ","End":"01:54.984","Text":"and this will be x squared."},{"Start":"01:54.984 ","End":"01:57.830","Text":"I better rewrite it, it\u0027s hard to see."},{"Start":"01:57.830 ","End":"02:04.850","Text":"It\u0027s the limit of x over x squared plus 1 as x goes to 0."},{"Start":"02:04.850 ","End":"02:08.460","Text":"Here there\u0027s no problem in substituting 0."},{"Start":"02:08.560 ","End":"02:14.300","Text":"0 over 0 squared plus 1 is just equal to 0."},{"Start":"02:14.300 ","End":"02:17.135","Text":"That\u0027s this limit."},{"Start":"02:17.135 ","End":"02:19.360","Text":"I\u0027ll just highlight it."},{"Start":"02:19.360 ","End":"02:21.435","Text":"Now let\u0027s try the other 1."},{"Start":"02:21.435 ","End":"02:23.330","Text":"I\u0027m going to tell you in advance, it\u0027s not going to work."},{"Start":"02:23.330 ","End":"02:24.560","Text":"It\u0027s going to give the same answer,"},{"Start":"02:24.560 ","End":"02:26.470","Text":"but we didn\u0027t know that."},{"Start":"02:26.470 ","End":"02:34.340","Text":"Let\u0027s try the limit as x goes to 0 along y equals 2x of the same thing,"},{"Start":"02:34.340 ","End":"02:36.860","Text":"x to the fourth y squared."},{"Start":"02:36.860 ","End":"02:42.585","Text":"This time we\u0027ll get the limit as x goes to 0. y is 2x,"},{"Start":"02:42.585 ","End":"02:51.040","Text":"so we get 2x cubed over x to the fourth"},{"Start":"02:59.870 ","End":"03:06.360","Text":"plus y squared is 4x squared."},{"Start":"03:06.360 ","End":"03:11.970","Text":"The same trick of dividing top and bottom by x squared here will"},{"Start":"03:11.970 ","End":"03:19.905","Text":"leave us with 2x"},{"Start":"03:19.905 ","End":"03:22.610","Text":"over x squared plus 4."},{"Start":"03:22.610 ","End":"03:27.515","Text":"But it\u0027s still going to be 0 over something non-zero."},{"Start":"03:27.515 ","End":"03:28.880","Text":"This is also equal."},{"Start":"03:28.880 ","End":"03:32.450","Text":"I didn\u0027t do all the steps to 0."},{"Start":"03:32.450 ","End":"03:35.560","Text":"I\u0027ll highlight this also."},{"Start":"03:35.560 ","End":"03:41.090","Text":"No good yet. So we have to try keep looking for paths until we get something different."},{"Start":"03:41.090 ","End":"03:45.260","Text":"There is a technique which I recommend."},{"Start":"03:45.260 ","End":"03:48.175","Text":"Often, it usually works."},{"Start":"03:48.175 ","End":"03:51.790","Text":"In case it\u0027s similar to this."},{"Start":"03:53.180 ","End":"03:56.150","Text":"It\u0027s a polynomial over a polynomial."},{"Start":"03:56.150 ","End":"04:02.825","Text":"But sometimes we can substitute y as a power of x in such a way"},{"Start":"04:02.825 ","End":"04:10.245","Text":"that we get all the powers of x to be the same on the top and on the bottom."},{"Start":"04:10.245 ","End":"04:12.710","Text":"Like here I have a power of 4 and a power of 2."},{"Start":"04:12.710 ","End":"04:16.775","Text":"Think about it, if I let y equal x squared,"},{"Start":"04:16.775 ","End":"04:19.775","Text":"then this will also be x to the fourth."},{"Start":"04:19.775 ","End":"04:21.500","Text":"So we\u0027ll have the same power."},{"Start":"04:21.500 ","End":"04:23.120","Text":"This is the idea."},{"Start":"04:23.120 ","End":"04:25.900","Text":"Now we\u0027re going to try the path,"},{"Start":"04:25.900 ","End":"04:27.825","Text":"maybe I\u0027ll even sketch it,"},{"Start":"04:27.825 ","End":"04:30.080","Text":"y equals x squared."},{"Start":"04:30.080 ","End":"04:35.105","Text":"I\u0027ll just label that y equals x squared and go along this path."},{"Start":"04:35.105 ","End":"04:38.275","Text":"Also, when x goes to 0, y goes to 0."},{"Start":"04:38.275 ","End":"04:40.840","Text":"This time we\u0027ll get the limit."},{"Start":"04:40.840 ","End":"04:50.690","Text":"x goes to 0. y equals x squared of x squared y over x to the fourth plus y squared."},{"Start":"04:50.690 ","End":"04:56.225","Text":"This time we\u0027ll get the limit as x goes to 0. x squared,"},{"Start":"04:56.225 ","End":"04:57.950","Text":"and then y is also x squared."},{"Start":"04:57.950 ","End":"05:00.319","Text":"So here we get x to the fourth,"},{"Start":"05:00.319 ","End":"05:03.680","Text":"and on the denominator we get x to the fourth."},{"Start":"05:03.680 ","End":"05:10.430","Text":"Like I said, y squared comes out to be x to the fourth, the same power."},{"Start":"05:10.430 ","End":"05:18.645","Text":"Now what happens is that this is equal to the limit as x goes to 0."},{"Start":"05:18.645 ","End":"05:22.020","Text":"This is x to the fourth over 2 x to the fourth."},{"Start":"05:22.020 ","End":"05:29.160","Text":"Or what I\u0027m saying is we could actually divide top and bottom by x to the fourth."},{"Start":"05:29.160 ","End":"05:32.870","Text":"This will be 1 and this will be 1."},{"Start":"05:33.080 ","End":"05:38.420","Text":"It\u0027s the limit as x goes to 0 of the constant 1 over 1 plus 1,"},{"Start":"05:38.420 ","End":"05:41.070","Text":"well, that\u0027s just a half."},{"Start":"05:41.530 ","End":"05:43.625","Text":"This time we did it,"},{"Start":"05:43.625 ","End":"05:46.045","Text":"1 half is different."},{"Start":"05:46.045 ","End":"05:50.300","Text":"After a minor setback where the green and the red paths came out the same,"},{"Start":"05:50.300 ","End":"05:54.065","Text":"the blue saved the day by giving us something different."},{"Start":"05:54.065 ","End":"05:56.750","Text":"Once we have 2 paths with different limits,"},{"Start":"05:56.750 ","End":"06:04.945","Text":"then we know that the original limit does not exist or is not existent,"},{"Start":"06:04.945 ","End":"06:06.170","Text":"or there is no limit,"},{"Start":"06:06.170 ","End":"06:09.390","Text":"and so on. We are done."}],"ID":8908},{"Watched":false,"Name":"Exercise 3 part f","Duration":"4m 41s","ChapterTopicVideoID":8559,"CourseChapterTopicPlaylistID":4970,"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 again have a function of 2 variables"},{"Start":"00:05.220 ","End":"00:11.085","Text":"and we have to check a double limit or a limit as x goes to 0 and y goes to 0."},{"Start":"00:11.085 ","End":"00:12.739","Text":"I don\u0027t know why in all the exercises,"},{"Start":"00:12.739 ","End":"00:15.800","Text":"it seems to be 0, 0, but there we are."},{"Start":"00:15.800 ","End":"00:17.455","Text":"We try the obvious,"},{"Start":"00:17.455 ","End":"00:20.670","Text":"which is to substitute x equals 0, y equals 0."},{"Start":"00:20.670 ","End":"00:22.380","Text":"That gives us 0 over 0,"},{"Start":"00:22.380 ","End":"00:24.000","Text":"so that\u0027s no good."},{"Start":"00:24.000 ","End":"00:26.775","Text":"Next, we try iterated limits."},{"Start":"00:26.775 ","End":"00:29.595","Text":"Well, 1 of the iterated limits doesn\u0027t exist."},{"Start":"00:29.595 ","End":"00:33.015","Text":"For example, if I let y equals 0,"},{"Start":"00:33.015 ","End":"00:36.915","Text":"then I get x cubed over 2x to the 6th,"},{"Start":"00:36.915 ","End":"00:39.140","Text":"and that doesn\u0027t have a limit when x goes to 0,"},{"Start":"00:39.140 ","End":"00:44.545","Text":"so we can\u0027t use the iterated limits."},{"Start":"00:44.545 ","End":"00:48.725","Text":"We try the next technique which is limit along a path."},{"Start":"00:48.725 ","End":"00:54.539","Text":"We try to find 2 paths such that the limits along each of the paths is different."},{"Start":"00:54.890 ","End":"01:01.085","Text":"A copied diagram from a previous exercise where we tried these 2 paths,"},{"Start":"01:01.085 ","End":"01:03.590","Text":"and it worked, but in another exercise,"},{"Start":"01:03.590 ","End":"01:06.080","Text":"they didn\u0027t work. Let\u0027s see."},{"Start":"01:06.080 ","End":"01:10.800","Text":"Let\u0027s try at least y equals x. I\u0027ll,"},{"Start":"01:10.800 ","End":"01:16.490","Text":"first of all, figure out the limit along y equals x."},{"Start":"01:16.490 ","End":"01:19.880","Text":"Of course, when x goes to 0, y goes to 0,"},{"Start":"01:19.880 ","End":"01:23.600","Text":"and what I get is the limit,"},{"Start":"01:23.600 ","End":"01:26.300","Text":"x goes to 0,"},{"Start":"01:26.300 ","End":"01:30.410","Text":"but y equals x of this thing here,"},{"Start":"01:30.410 ","End":"01:37.700","Text":"x cubed y over 2x to the 6th plus y squared."},{"Start":"01:37.700 ","End":"01:40.280","Text":"What this equals is limit."},{"Start":"01:40.280 ","End":"01:42.049","Text":"We let x go to 0,"},{"Start":"01:42.049 ","End":"01:45.115","Text":"but we substitute y whatever it equals,"},{"Start":"01:45.115 ","End":"01:47.340","Text":"in this case, x,"},{"Start":"01:47.340 ","End":"01:53.175","Text":"so we get x cubed times x is x to the 4th,"},{"Start":"01:53.175 ","End":"01:57.015","Text":"on the denominator, 2x to the 6th,"},{"Start":"01:57.015 ","End":"02:01.390","Text":"and y is x, so it\u0027s plus x squared."},{"Start":"02:01.390 ","End":"02:05.929","Text":"If we divide top and bottom by x squared,"},{"Start":"02:05.929 ","End":"02:10.085","Text":"which is the lowest but the highest we can divide by,"},{"Start":"02:10.085 ","End":"02:13.190","Text":"we get the limit as x goes to 0."},{"Start":"02:13.190 ","End":"02:19.500","Text":"Let\u0027s see. X squared over 2x to the 4th plus 1,"},{"Start":"02:19.500 ","End":"02:21.725","Text":"canceling by x squared everything."},{"Start":"02:21.725 ","End":"02:24.800","Text":"At this point, we can substitute x equals 0,"},{"Start":"02:24.800 ","End":"02:27.475","Text":"and here, we get 0 over 0 plus 1."},{"Start":"02:27.475 ","End":"02:34.380","Text":"This is just 0 and I\u0027ll mark it."},{"Start":"02:34.380 ","End":"02:36.935","Text":"Now, in the previous exercise,"},{"Start":"02:36.935 ","End":"02:41.210","Text":"we had something similar to this and we tried the limit along this and we also got 0."},{"Start":"02:41.210 ","End":"02:42.980","Text":"I\u0027m going to save time and say,"},{"Start":"02:42.980 ","End":"02:44.210","Text":"this is not going to work,"},{"Start":"02:44.210 ","End":"02:45.410","Text":"it\u0027s also going to give 0,"},{"Start":"02:45.410 ","End":"02:47.140","Text":"I\u0027m going to erase it."},{"Start":"02:47.140 ","End":"02:51.470","Text":"I gave a hint in the previous exercise, but in case you missed it,"},{"Start":"02:51.470 ","End":"02:57.440","Text":"the idea is to let y equal a power of x so we get the same power of x."},{"Start":"02:57.440 ","End":"02:58.770","Text":"Now here, I have x to the sixth,"},{"Start":"02:58.770 ","End":"03:00.485","Text":"and here, I have y squared."},{"Start":"03:00.485 ","End":"03:03.635","Text":"If I let y equal x cubed,"},{"Start":"03:03.635 ","End":"03:06.095","Text":"in fact, I\u0027m going to even draw that already,"},{"Start":"03:06.095 ","End":"03:08.620","Text":"y equals x cubed,"},{"Start":"03:08.620 ","End":"03:12.120","Text":"I\u0027ll just mark that y equals x cubed,"},{"Start":"03:12.120 ","End":"03:14.400","Text":"and also x goes to 0,"},{"Start":"03:14.400 ","End":"03:17.145","Text":"y will also go to 0."},{"Start":"03:17.145 ","End":"03:20.475","Text":"Then we\u0027ll get x to the 6th and x to the 6th."},{"Start":"03:20.475 ","End":"03:24.940","Text":"In fact, you\u0027ll also get x to the 6th on the top, which is great."},{"Start":"03:24.950 ","End":"03:33.125","Text":"Now, we\u0027ll take the limit along the path y equals x cubed as x goes to 0."},{"Start":"03:33.125 ","End":"03:39.980","Text":"In other words, the limit as x goes to 0 along y equals x cubed of the same thing,"},{"Start":"03:39.980 ","End":"03:46.110","Text":"x cubed y over 2x to the 6th plus y squared."},{"Start":"03:46.110 ","End":"03:49.500","Text":"This is the limit as x goes to 0,"},{"Start":"03:49.500 ","End":"03:51.180","Text":"I let y be x cubed."},{"Start":"03:51.180 ","End":"03:52.320","Text":"Here, I\u0027ve got x cubed,"},{"Start":"03:52.320 ","End":"03:55.375","Text":"x cubed, which is x to the 6th."},{"Start":"03:55.375 ","End":"04:01.770","Text":"Here, I have 2x to the 6th and y squared is x to the 6th."},{"Start":"04:01.770 ","End":"04:06.980","Text":"Now, the x to the 6th cancels and so what I get is"},{"Start":"04:06.980 ","End":"04:13.875","Text":"the limit of 1 over 2 plus 1 as x goes to 0."},{"Start":"04:13.875 ","End":"04:15.380","Text":"Well, it\u0027s a constant to the 3rd,"},{"Start":"04:15.380 ","End":"04:18.020","Text":"so the limit is also a 3rd."},{"Start":"04:18.020 ","End":"04:23.300","Text":"I\u0027ll highlight it and this is good for us because it\u0027s different."},{"Start":"04:23.300 ","End":"04:25.430","Text":"Because these 2 are different,"},{"Start":"04:25.430 ","End":"04:27.765","Text":"different limits along different paths,"},{"Start":"04:27.765 ","End":"04:30.600","Text":"we conclude that this limit doesn\u0027t exist or this"},{"Start":"04:30.600 ","End":"04:34.050","Text":"has no limit or whatever you want to phrase it,"},{"Start":"04:34.050 ","End":"04:37.030","Text":"but let\u0027s just say has no limit,"},{"Start":"04:37.030 ","End":"04:41.120","Text":"and nothing we can do, so we\u0027re done."}],"ID":8909},{"Watched":false,"Name":"Exercise 3 part g","Duration":"6m 38s","ChapterTopicVideoID":8560,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.795","Text":"In this exercise, we have to try and compute the following limit equals it may not exist."},{"Start":"00:06.795 ","End":"00:09.645","Text":"Why is there even a problem?"},{"Start":"00:09.645 ","End":"00:14.250","Text":"The first thing we do is try substituting x equals 0 and y equals 0."},{"Start":"00:14.250 ","End":"00:17.160","Text":"But if we do that, we get 0 over 0."},{"Start":"00:17.160 ","End":"00:18.884","Text":"That\u0027s why there\u0027s a problem."},{"Start":"00:18.884 ","End":"00:23.790","Text":"Usually the first thing to try if you suspect there is no limit,"},{"Start":"00:23.790 ","End":"00:28.140","Text":"is the iterated limit test where we let first x go"},{"Start":"00:28.140 ","End":"00:32.620","Text":"to 0 and then y go to 0 and vice versa."},{"Start":"00:32.620 ","End":"00:36.860","Text":"But it turns out that both iterated limits are equal to 0."},{"Start":"00:36.860 ","End":"00:39.710","Text":"For example, if I let x equals 0,"},{"Start":"00:39.710 ","End":"00:46.650","Text":"then the numerator is 0 and I get 0 over y squared, which is 0."},{"Start":"00:46.650 ","End":"00:49.535","Text":"The limit as y goes to 0 of 0 is 0,"},{"Start":"00:49.535 ","End":"00:51.260","Text":"and the same thing the other way around."},{"Start":"00:51.260 ","End":"00:53.135","Text":"The iterated limits exist,"},{"Start":"00:53.135 ","End":"00:55.675","Text":"but they\u0027re equal, so that\u0027s no help."},{"Start":"00:55.675 ","End":"01:00.335","Text":"The last trick we know is limits along a path."},{"Start":"01:00.335 ","End":"01:04.100","Text":"Or rather, we try to look for 2 different paths and hope"},{"Start":"01:04.100 ","End":"01:08.605","Text":"to get different limits along different paths."},{"Start":"01:08.605 ","End":"01:12.005","Text":"I just wrote the concept limits along the path."},{"Start":"01:12.005 ","End":"01:14.270","Text":"We\u0027re going to look for 2 paths."},{"Start":"01:14.270 ","End":"01:16.775","Text":"Now, as in the previous exercise,"},{"Start":"01:16.775 ","End":"01:20.210","Text":"often at least 1 of the things to try is to let y"},{"Start":"01:20.210 ","End":"01:24.064","Text":"be a power of x in such a way that we get the same powers."},{"Start":"01:24.064 ","End":"01:28.895","Text":"I remember we once had y equals x squared then we once had y equals x cubed and so on."},{"Start":"01:28.895 ","End":"01:31.850","Text":"Here, if we just let y equals x,"},{"Start":"01:31.850 ","End":"01:35.610","Text":"would be 1 path."},{"Start":"01:35.610 ","End":"01:39.185","Text":"Then we would get x squared here plus x squared here,"},{"Start":"01:39.185 ","End":"01:41.450","Text":"and would even get x squared here."},{"Start":"01:41.450 ","End":"01:44.660","Text":"All same powers of x sounds good."},{"Start":"01:44.660 ","End":"01:51.590","Text":"In fact, 1 thing to do is to try different constant times x."},{"Start":"01:51.590 ","End":"01:55.700","Text":"For example, we might try y equals x and y equals 2x,"},{"Start":"01:55.700 ","End":"01:58.790","Text":"because 2x would also give us the same powers of x."},{"Start":"01:58.790 ","End":"02:01.305","Text":"We\u0027d have x squared and we would have 4x squared,"},{"Start":"02:01.305 ","End":"02:02.955","Text":"but we still have all x squared."},{"Start":"02:02.955 ","End":"02:04.650","Text":"Here we\u0027d have 2x squared,"},{"Start":"02:04.650 ","End":"02:06.060","Text":"so everything with x squared."},{"Start":"02:06.060 ","End":"02:07.860","Text":"In fact, let\u0027s try these 2."},{"Start":"02:07.860 ","End":"02:11.425","Text":"I kept a sketch from a previous exercise."},{"Start":"02:11.425 ","End":"02:14.750","Text":"Here it is. It doesn\u0027t really help to have a sketch,"},{"Start":"02:14.750 ","End":"02:16.850","Text":"but it\u0027s nice to look at."},{"Start":"02:16.850 ","End":"02:20.345","Text":"For some reason in all these exercises."},{"Start":"02:20.345 ","End":"02:23.450","Text":"I brought x goes to 0, y goes to 0,"},{"Start":"02:23.450 ","End":"02:28.240","Text":"but I should remind you it could be any 2 numbers, a and b."},{"Start":"02:28.240 ","End":"02:31.580","Text":"Let\u0027s take this one is the red one,"},{"Start":"02:31.580 ","End":"02:32.900","Text":"this one\u0027s the green one."},{"Start":"02:32.900 ","End":"02:34.880","Text":"Let\u0027s do the red one first."},{"Start":"02:34.880 ","End":"02:42.305","Text":"We\u0027ll have the limit as x goes to 0 along the path y equals x"},{"Start":"02:42.305 ","End":"02:50.179","Text":"of sine xy over x squared plus y squared."},{"Start":"02:50.179 ","End":"02:53.135","Text":"This equals the limit."},{"Start":"02:53.135 ","End":"02:57.905","Text":"Just x goes to 0 and we substitute what y is."},{"Start":"02:57.905 ","End":"03:02.030","Text":"Y is equal to x so we get sine of x times x,"},{"Start":"03:02.030 ","End":"03:06.320","Text":"which is x squared over x squared plus x squared,"},{"Start":"03:06.320 ","End":"03:10.530","Text":"which is 2x squared as x goes to 0."},{"Start":"03:10.530 ","End":"03:14.170","Text":"Now, there\u0027s actually a famous limit which I\u0027m going to"},{"Start":"03:14.170 ","End":"03:17.410","Text":"use here and we\u0027ve seen it many times and I\u0027ll write it over here."},{"Start":"03:17.410 ","End":"03:18.700","Text":"The limit, let\u0027s see,"},{"Start":"03:18.700 ","End":"03:20.770","Text":"I\u0027ll choose the variable Alpha."},{"Start":"03:20.770 ","End":"03:23.005","Text":"When Alpha goes to 0,"},{"Start":"03:23.005 ","End":"03:27.420","Text":"sine of Alpha over Alpha,"},{"Start":"03:27.420 ","End":"03:29.460","Text":"this limit equals 1,"},{"Start":"03:29.460 ","End":"03:31.185","Text":"it goes to 1."},{"Start":"03:31.185 ","End":"03:35.370","Text":"Here if I just let Alpha equals x squared,"},{"Start":"03:35.370 ","End":"03:38.110","Text":"say I let Alpha equals x squared."},{"Start":"03:38.110 ","End":"03:39.520","Text":"Then when x goes to 0 of x,"},{"Start":"03:39.520 ","End":"03:44.020","Text":"but also goes to 0, so I get and I can take the 1/2 out front,"},{"Start":"03:44.020 ","End":"03:53.595","Text":"a half limit as Alpha goes to 0 of sine Alpha over Alpha."},{"Start":"03:53.595 ","End":"03:58.140","Text":"Since this is 1/2 times 1,"},{"Start":"03:58.140 ","End":"04:03.240","Text":"this gives us 1/2."},{"Start":"04:03.240 ","End":"04:07.670","Text":"I\u0027ll highlight it for comparison with the next path."},{"Start":"04:07.670 ","End":"04:10.940","Text":"I\u0027ll just note, of course you could have done this with L\u0027Hopital if you"},{"Start":"04:10.940 ","End":"04:14.720","Text":"didn\u0027t see the sine Alpha over Alpha thing,"},{"Start":"04:14.720 ","End":"04:16.865","Text":"and you would have got the same answer."},{"Start":"04:16.865 ","End":"04:19.940","Text":"Now let\u0027s take the other path where y is 2x."},{"Start":"04:19.940 ","End":"04:28.490","Text":"We get the limit as x goes to 0 along y equals 2x of sine."},{"Start":"04:28.490 ","End":"04:33.785","Text":"Same thing, xy over x squared plus y squared."},{"Start":"04:33.785 ","End":"04:37.870","Text":"This time, if we let y equals 2x,"},{"Start":"04:37.870 ","End":"04:41.929","Text":"we get the limit as x goes to 0."},{"Start":"04:41.929 ","End":"04:45.680","Text":"Here we get sine of y is 2x,"},{"Start":"04:45.680 ","End":"04:51.390","Text":"so it\u0027s 2x squared over."},{"Start":"04:51.980 ","End":"04:57.600","Text":"On the denominator we get x squared plus 2x."},{"Start":"04:57.600 ","End":"04:59.355","Text":"All squared is 4x squared,"},{"Start":"04:59.355 ","End":"05:04.000","Text":"a total of 5x squared."},{"Start":"05:05.020 ","End":"05:09.620","Text":"Once again, I\u0027d rather use this limit than use L\u0027Hopital all though it\u0027s up"},{"Start":"05:09.620 ","End":"05:13.550","Text":"to you. Here\u0027s the trick."},{"Start":"05:13.550 ","End":"05:18.960","Text":"Instead of the 5, I\u0027m going to write a 2, but that\u0027s cheating."},{"Start":"05:18.960 ","End":"05:23.430","Text":"What I\u0027m going to do is multiply here by 2,"},{"Start":"05:23.430 ","End":"05:26.905","Text":"that will cancel with this 2 and restore the original 5."},{"Start":"05:26.905 ","End":"05:29.350","Text":"If you multiply this out, the 2 cancels and we get"},{"Start":"05:29.350 ","End":"05:33.530","Text":"5x squared just like it was supposed to be."},{"Start":"05:34.050 ","End":"05:37.510","Text":"Maybe just for the record to say originally we had"},{"Start":"05:37.510 ","End":"05:40.210","Text":"5x squared here and we didn\u0027t have this."},{"Start":"05:40.210 ","End":"05:44.695","Text":"But then we multiply it by 2 over 5 and put a 2 here."},{"Start":"05:44.695 ","End":"05:47.785","Text":"That\u0027s okay to say that was what it was."},{"Start":"05:47.785 ","End":"05:49.330","Text":"Then, okay."},{"Start":"05:49.330 ","End":"05:55.840","Text":"Now we can let Alpha equals 2x squared."},{"Start":"05:56.120 ","End":"05:58.920","Text":"This thing is a constant,"},{"Start":"05:58.920 ","End":"06:02.950","Text":"so we get 2/5."},{"Start":"06:03.410 ","End":"06:12.150","Text":"The limit as Alpha goes to 0 of sine Alpha over Alpha."},{"Start":"06:12.150 ","End":"06:16.695","Text":"That\u0027s equal to 1 times 2/5."},{"Start":"06:16.695 ","End":"06:22.245","Text":"I\u0027ll write it over here. We get that this is equal to 2/5."},{"Start":"06:22.245 ","End":"06:26.320","Text":"Since 2/5 is not equal to 1/2,"},{"Start":"06:26.320 ","End":"06:31.910","Text":"it means that this thing has no limit,"},{"Start":"06:31.910 ","End":"06:33.185","Text":"doesn\u0027t have a limit,"},{"Start":"06:33.185 ","End":"06:39.480","Text":"limit does not exist or something like that and we\u0027re done."}],"ID":8910},{"Watched":false,"Name":"Exercise 3 part h","Duration":"9m 50s","ChapterTopicVideoID":8561,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.560","Text":"In this exercise, we have a limit of a function or an expression of 3 variables,"},{"Start":"00:07.560 ","End":"00:09.180","Text":"x, y, and z."},{"Start":"00:09.180 ","End":"00:13.980","Text":"We didn\u0027t really discuss this much if at all in the tutorial,"},{"Start":"00:13.980 ","End":"00:17.070","Text":"but it\u0027s very similar to 2 variables."},{"Start":"00:17.070 ","End":"00:19.630","Text":"Most of the things carry over."},{"Start":"00:20.660 ","End":"00:27.660","Text":"The first thing we would try in such a limit would be to just substitute."},{"Start":"00:27.660 ","End":"00:29.520","Text":"We try x equals 0,"},{"Start":"00:29.520 ","End":"00:31.425","Text":"y equals 0, z equals 0."},{"Start":"00:31.425 ","End":"00:35.535","Text":"We see that we get 0 in the numerator and in the denominator."},{"Start":"00:35.535 ","End":"00:37.560","Text":"We have a 0/0 situation,"},{"Start":"00:37.560 ","End":"00:40.540","Text":"so it\u0027s not so straightforward."},{"Start":"00:40.540 ","End":"00:46.730","Text":"Now, the iterated limits method also works in 3D,"},{"Start":"00:46.730 ","End":"00:51.100","Text":"but unfortunately, all the iterated limits are equal."},{"Start":"00:51.100 ","End":"00:53.975","Text":"If I wanted to prove that this limit doesn\u0027t exist,"},{"Start":"00:53.975 ","End":"00:55.475","Text":"I can\u0027t use iterated limits."},{"Start":"00:55.475 ","End":"00:57.065","Text":"For example, if I,"},{"Start":"00:57.065 ","End":"01:00.200","Text":"first of all took the limit as x goes to 0,"},{"Start":"01:00.200 ","End":"01:08.105","Text":"then I\u0027d get 0 over y to the fourth plus z to the fourth,"},{"Start":"01:08.105 ","End":"01:10.985","Text":"and that\u0027s just 0, so the limit would be 0."},{"Start":"01:10.985 ","End":"01:13.234","Text":"Likewise with all the variables,"},{"Start":"01:13.234 ","End":"01:15.620","Text":"I\u0027m going to get all the 3 iterated, well,"},{"Start":"01:15.620 ","End":"01:17.870","Text":"there is more than 3 iterated limits, there\u0027s actually 6."},{"Start":"01:17.870 ","End":"01:20.120","Text":"I mean, I can take x,y,z,"},{"Start":"01:20.120 ","End":"01:23.655","Text":"x,z,y, and so on, all permutations."},{"Start":"01:23.655 ","End":"01:27.735","Text":"But all the iterated limits are equal, that\u0027s not good."},{"Start":"01:27.735 ","End":"01:33.425","Text":"The other technique we learned in 2D is the method of trying different paths."},{"Start":"01:33.425 ","End":"01:35.540","Text":"This is what we\u0027re going to use here."},{"Start":"01:35.540 ","End":"01:39.549","Text":"Only the paths are going to be paths in 3D space."},{"Start":"01:39.549 ","End":"01:42.120","Text":"It\u0027s hard to sketch in 3D,"},{"Start":"01:42.120 ","End":"01:43.395","Text":"so I won\u0027t even try."},{"Start":"01:43.395 ","End":"01:51.120","Text":"I\u0027ll just show you an example of a path that goes to 0,0,0."},{"Start":"01:51.190 ","End":"01:58.480","Text":"One very easy example of a path that goes through the origin,"},{"Start":"01:58.480 ","End":"02:02.615","Text":"and I\u0027ll use parametric form this time,"},{"Start":"02:02.615 ","End":"02:08.270","Text":"would be just to take x equals t,"},{"Start":"02:08.270 ","End":"02:18.080","Text":"y equals t, z equals t. Then when t goes to 0, easily,"},{"Start":"02:18.080 ","End":"02:21.845","Text":"we can see that x, y,"},{"Start":"02:21.845 ","End":"02:26.290","Text":"z goes to 0,0,0,"},{"Start":"02:26.290 ","End":"02:28.490","Text":"or perhaps I should have written it in this form,"},{"Start":"02:28.490 ","End":"02:32.320","Text":"x goes to 0, y goes to 0, z goes to 0."},{"Start":"02:32.320 ","End":"02:37.410","Text":"It\u0027s actually a straight line that\u0027s symmetrical about all the axes."},{"Start":"02:37.410 ","End":"02:40.625","Text":"It\u0027s smack in the middle of the first octant."},{"Start":"02:40.625 ","End":"02:45.405","Text":"Anyway, it\u0027s a straight line and it goes through the origin."},{"Start":"02:45.405 ","End":"02:55.320","Text":"What we could do now is to calculate this limit as the limit as t goes to 0."},{"Start":"02:56.000 ","End":"02:59.380","Text":"I guess I\u0027ll also write,"},{"Start":"02:59.680 ","End":"03:03.410","Text":"not sure exactly how to write it without being clumsy."},{"Start":"03:03.410 ","End":"03:09.540","Text":"Maybe I\u0027ll write it as x,y,z."},{"Start":"03:09.770 ","End":"03:12.780","Text":"Vectors even do it in vector."},{"Start":"03:12.780 ","End":"03:14.520","Text":"I need to write this here."},{"Start":"03:14.520 ","End":"03:19.080","Text":"Let\u0027s just write it just more"},{"Start":"03:19.080 ","End":"03:25.850","Text":"compactly as x,y,z equals t,t,t."},{"Start":"03:25.850 ","End":"03:34.129","Text":"Then we copy this x,y,z over x squared plus y to the fourth,"},{"Start":"03:34.129 ","End":"03:36.965","Text":"plus z to the fourth."},{"Start":"03:36.965 ","End":"03:44.490","Text":"What we get is the limit as t goes to 0,"},{"Start":"03:44.490 ","End":"03:49.085","Text":"t times t times t is t cubed."},{"Start":"03:49.085 ","End":"03:53.585","Text":"Here we have t times t is t squared."},{"Start":"03:53.585 ","End":"03:56.630","Text":"Here, t to the fourth and here t to the fourth,"},{"Start":"03:56.630 ","End":"04:00.695","Text":"so 2t to the fourth."},{"Start":"04:00.695 ","End":"04:05.090","Text":"Now it\u0027s a limit in just 1 variable."},{"Start":"04:05.090 ","End":"04:10.290","Text":"I just like to point out that we don\u0027t have to use the parameter t,"},{"Start":"04:10.290 ","End":"04:12.560","Text":"we could have just used x,"},{"Start":"04:12.560 ","End":"04:15.080","Text":"similar to that we did in 2 variables,"},{"Start":"04:15.080 ","End":"04:18.080","Text":"only this time we would need 2 functions."},{"Start":"04:18.080 ","End":"04:24.485","Text":"We would say like the limit as x goes to 0 and y equals"},{"Start":"04:24.485 ","End":"04:33.010","Text":"x and z equals x or in general 2 functions of x here of the same thing,"},{"Start":"04:34.700 ","End":"04:43.955","Text":"and this would equal this only with x. I just copied this and replaced t with x."},{"Start":"04:43.955 ","End":"04:50.435","Text":"But I don\u0027t see why you wouldn\u0027t want to use a parameter, it\u0027s pretty flexible."},{"Start":"04:50.435 ","End":"04:53.005","Text":"I\u0027ll just stick with the parameter form,"},{"Start":"04:53.005 ","End":"04:54.855","Text":"just saying there\u0027s no alternative,"},{"Start":"04:54.855 ","End":"04:59.420","Text":"and now we just compute the limit."},{"Start":"04:59.420 ","End":"05:04.155","Text":"We can certainly divide top and bottom by t squared,"},{"Start":"05:04.155 ","End":"05:06.980","Text":"and that will give us the limit,"},{"Start":"05:06.980 ","End":"05:08.240","Text":"t goes to 0."},{"Start":"05:08.240 ","End":"05:10.010","Text":"Now t goes to 0, so t is not 0,"},{"Start":"05:10.010 ","End":"05:11.615","Text":"so t squared is not 0."},{"Start":"05:11.615 ","End":"05:14.660","Text":"Dividing by t squared here we get t, here,"},{"Start":"05:14.660 ","End":"05:19.080","Text":"we get 1, here we get 2t squared."},{"Start":"05:19.080 ","End":"05:25.480","Text":"When t goes to 0, we get 0/1,"},{"Start":"05:25.480 ","End":"05:28.295","Text":"and 0/1 is just 0."},{"Start":"05:28.295 ","End":"05:30.290","Text":"I\u0027ll highlight it."},{"Start":"05:30.290 ","End":"05:34.729","Text":"Now we need to look for another path which won\u0027t give us 0."},{"Start":"05:34.729 ","End":"05:37.790","Text":"For the other path, I\u0027m going to use the idea,"},{"Start":"05:37.790 ","End":"05:41.060","Text":"the concept we used in 2 variables in"},{"Start":"05:41.060 ","End":"05:46.265","Text":"similar situations where I said we\u0027d like to get everything to be with the same power."},{"Start":"05:46.265 ","End":"05:50.700","Text":"If I want y and z to be functions of x,"},{"Start":"05:51.110 ","End":"05:54.210","Text":"to get the same power,"},{"Start":"05:54.210 ","End":"05:55.995","Text":"the x squared stays."},{"Start":"05:55.995 ","End":"05:59.090","Text":"If I let y equals the square root of x,"},{"Start":"05:59.090 ","End":"06:04.890","Text":"then y to the fourth will be x squared and similarly for z."},{"Start":"06:05.240 ","End":"06:07.520","Text":"What I\u0027m saying is,"},{"Start":"06:07.520 ","End":"06:11.075","Text":"is that in parametric form, well,"},{"Start":"06:11.075 ","End":"06:12.320","Text":"in non parametric form,"},{"Start":"06:12.320 ","End":"06:16.700","Text":"I just said y equals the square root of x and z equals square root of x."},{"Start":"06:16.700 ","End":"06:22.730","Text":"But I\u0027d like to use parameters at least one time so you can see."},{"Start":"06:22.730 ","End":"06:26.915","Text":"We just let x equal t. I mean,"},{"Start":"06:26.915 ","End":"06:30.305","Text":"just like another name everywhere is x I use t,"},{"Start":"06:30.305 ","End":"06:32.090","Text":"then y is, well,"},{"Start":"06:32.090 ","End":"06:35.000","Text":"instead of square root of x, I\u0027ll put square root of t and"},{"Start":"06:35.000 ","End":"06:39.480","Text":"z equals square root of t and this is like scratch,"},{"Start":"06:39.480 ","End":"06:42.535","Text":"I\u0027ll just erase it."},{"Start":"06:42.535 ","End":"06:45.495","Text":"I think I\u0027ll move this down."},{"Start":"06:45.495 ","End":"06:49.290","Text":"We see that when t goes to 0,"},{"Start":"06:49.290 ","End":"06:54.074","Text":"then similarly as before,"},{"Start":"06:54.074 ","End":"07:01.815","Text":"x,y,z goes to 0,0,0, an arrow here."},{"Start":"07:01.815 ","End":"07:07.070","Text":"Technical point, because the square root is only defined by positives,"},{"Start":"07:07.070 ","End":"07:10.760","Text":"we better make it t going to 0 from the right. It\u0027s still a path."},{"Start":"07:10.760 ","End":"07:14.630","Text":"I\u0027m just going from the right to that point."},{"Start":"07:14.630 ","End":"07:19.555","Text":"Going from the left would be another path and then everything\u0027s fine."},{"Start":"07:19.555 ","End":"07:24.870","Text":"Let\u0027s see what the limit comes out."},{"Start":"07:24.870 ","End":"07:33.330","Text":"We get the limit as t goes to 0."},{"Start":"07:33.330 ","End":"07:43.335","Text":"I\u0027ll just make a note that x,y,z is equal to t root t. Root t,"},{"Start":"07:43.335 ","End":"07:50.410","Text":"rather than writing 3 separate lines of this thing,"},{"Start":"07:50.780 ","End":"07:55.220","Text":"x,y,z over x squared plus y to the fourth,"},{"Start":"07:55.220 ","End":"07:57.560","Text":"plus z to the fourth,"},{"Start":"07:57.560 ","End":"08:00.655","Text":"then it\u0027s going to equal the limit,"},{"Start":"08:00.655 ","End":"08:04.720","Text":"t goes to 0, one of the substitute."},{"Start":"08:05.960 ","End":"08:08.520","Text":"x,y,z is going to be, well,"},{"Start":"08:08.520 ","End":"08:10.755","Text":"I\u0027ll spell it out, it\u0027s t root t,"},{"Start":"08:10.755 ","End":"08:17.190","Text":"root t. Here I have x squared is t squared,"},{"Start":"08:17.190 ","End":"08:21.740","Text":"but y to the fourth is root t to the fourth,"},{"Start":"08:21.740 ","End":"08:26.505","Text":"which is actually t squared."},{"Start":"08:26.505 ","End":"08:31.275","Text":"Similarly, root t to the fourth is t squared."},{"Start":"08:31.275 ","End":"08:34.950","Text":"I need some more space here."},{"Start":"08:34.950 ","End":"08:38.550","Text":"This is equal to limit,"},{"Start":"08:38.550 ","End":"08:39.840","Text":"t goes to 0."},{"Start":"08:39.840 ","End":"08:41.850","Text":"Here I have root t, root t, is t,"},{"Start":"08:41.850 ","End":"08:47.085","Text":"t times t is t squared over 3t squared."},{"Start":"08:47.085 ","End":"08:52.360","Text":"Of course I can cancel the t squared."},{"Start":"08:52.550 ","End":"08:56.360","Text":"Now limit as t goes to 0 of a third,"},{"Start":"08:56.360 ","End":"08:59.060","Text":"which is a constant, is just that constant."},{"Start":"08:59.060 ","End":"09:00.670","Text":"It\u0027s a third."},{"Start":"09:00.670 ","End":"09:06.410","Text":"I\u0027ll highlight it, and now we definitely see that we got 2 different results."},{"Start":"09:06.410 ","End":"09:15.690","Text":"For this path, x equals t,"},{"Start":"09:15.690 ","End":"09:17.025","Text":"y equals t, z equals t,"},{"Start":"09:17.025 ","End":"09:18.674","Text":"we get this answer,"},{"Start":"09:18.674 ","End":"09:20.360","Text":"and for this path,"},{"Start":"09:20.360 ","End":"09:23.735","Text":"the second path, we get a different answer."},{"Start":"09:23.735 ","End":"09:27.980","Text":"Because we get 2 different answers for 2 different paths, as before,"},{"Start":"09:27.980 ","End":"09:34.355","Text":"I can go and claim that the limit here,"},{"Start":"09:34.355 ","End":"09:40.055","Text":"this limit just does not exist."},{"Start":"09:40.055 ","End":"09:46.100","Text":"There is no limit and that\u0027s our conclusion."},{"Start":"09:46.100 ","End":"09:50.340","Text":"We can\u0027t compute it, it doesn\u0027t exist. We\u0027re done."}],"ID":8911},{"Watched":false,"Name":"Exercise 4 part a","Duration":"3m 40s","ChapterTopicVideoID":8562,"CourseChapterTopicPlaylistID":4970,"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 compute the following limit."},{"Start":"00:03.900 ","End":"00:05.670","Text":"It\u0027s in 2 variables,"},{"Start":"00:05.670 ","End":"00:10.125","Text":"x and y both go to 0 of this expression."},{"Start":"00:10.125 ","End":"00:17.520","Text":"The immediate obstacle if we try to substitute is that we get 0 over 0."},{"Start":"00:17.520 ","End":"00:19.020","Text":"If you let x and y both be 0,"},{"Start":"00:19.020 ","End":"00:20.580","Text":"we get 0 over 0."},{"Start":"00:20.580 ","End":"00:21.960","Text":"We see there\u0027s a problem."},{"Start":"00:21.960 ","End":"00:25.320","Text":"It isn\u0027t just straightforward plugging in the numbers."},{"Start":"00:25.320 ","End":"00:29.250","Text":"There\u0027s not that many techniques for 2 variables."},{"Start":"00:29.250 ","End":"00:33.680","Text":"We have basically 2 techniques for trying"},{"Start":"00:33.680 ","End":"00:38.225","Text":"to show what the limit is and 2 techniques for proving that there isn\u0027t a limit,"},{"Start":"00:38.225 ","End":"00:44.750","Text":"and the only thing that might help here would be polar substitution,"},{"Start":"00:44.750 ","End":"00:47.585","Text":"so let\u0027s try that."},{"Start":"00:47.585 ","End":"00:50.615","Text":"I\u0027ve tried it and that\u0027s the 1 to use."},{"Start":"00:50.615 ","End":"00:52.970","Text":"But you might not know in advance or you"},{"Start":"00:52.970 ","End":"00:55.895","Text":"just try the various techniques that you\u0027ve learned."},{"Start":"00:55.895 ","End":"00:59.120","Text":"Now, to remind you what polar substitution is,"},{"Start":"00:59.120 ","End":"01:04.144","Text":"we let x equal r cosine Theta,"},{"Start":"01:04.144 ","End":"01:07.925","Text":"and y equals r sine Theta."},{"Start":"01:07.925 ","End":"01:10.235","Text":"Basically, it\u0027s polar coordinates,"},{"Start":"01:10.235 ","End":"01:12.770","Text":"and when we have x,"},{"Start":"01:12.770 ","End":"01:14.630","Text":"y going to 0,0,"},{"Start":"01:14.630 ","End":"01:16.820","Text":"it means that r goes to 0."},{"Start":"01:16.820 ","End":"01:23.880","Text":"Well, strictly speaking, 0 from positive values or from the right."},{"Start":"01:25.390 ","End":"01:30.050","Text":"The other useful thing to remember with polar is that x squared"},{"Start":"01:30.050 ","End":"01:34.510","Text":"plus y squared is equal to r-squared."},{"Start":"01:34.510 ","End":"01:36.770","Text":"I\u0027m not going to derive it again,"},{"Start":"01:36.770 ","End":"01:38.480","Text":"but just to remind you why,"},{"Start":"01:38.480 ","End":"01:40.640","Text":"cosine squared plus sine squared is 1."},{"Start":"01:40.640 ","End":"01:43.250","Text":"If you compute this, that\u0027s what you get."},{"Start":"01:43.250 ","End":"01:45.890","Text":"Now, let\u0027s make the substitution,"},{"Start":"01:45.890 ","End":"01:48.304","Text":"and so we get the limit."},{"Start":"01:48.304 ","End":"01:56.010","Text":"As r goes to positive 0 of, let see,"},{"Start":"01:56.010 ","End":"02:02.840","Text":"x cubed is r cubed cosine cubed Theta,"},{"Start":"02:02.840 ","End":"02:08.450","Text":"and y is r sine Theta, and as I said,"},{"Start":"02:08.450 ","End":"02:12.085","Text":"x squared plus y squared is r squared,"},{"Start":"02:12.085 ","End":"02:15.539","Text":"and we can simplify this."},{"Start":"02:15.539 ","End":"02:18.560","Text":"Basically, what we have here is r cubed times r"},{"Start":"02:18.560 ","End":"02:22.865","Text":"is r^4 and r^4 over r squared is r squared."},{"Start":"02:22.865 ","End":"02:29.305","Text":"We have the limit of r squared and then,"},{"Start":"02:29.305 ","End":"02:34.990","Text":"cosine cubed Theta sine Theta."},{"Start":"02:34.990 ","End":"02:37.610","Text":"Now, let\u0027s see what this equals."},{"Start":"02:37.610 ","End":"02:39.920","Text":"Now, what I\u0027m claiming is we have here"},{"Start":"02:39.920 ","End":"02:44.599","Text":"a situation where we have something, it tends to 0."},{"Start":"02:44.599 ","End":"02:48.535","Text":"This certainly tends to 0 when r goes to 0,"},{"Start":"02:48.535 ","End":"02:52.910","Text":"and this is bounded."},{"Start":"02:53.350 ","End":"02:55.580","Text":"Why is this bounded?"},{"Start":"02:55.580 ","End":"03:01.565","Text":"Well, sine and cosine of each of them are between minus 1, and 1."},{"Start":"03:01.565 ","End":"03:07.444","Text":"Cosine Theta in absolute value is less than or equal to 1,"},{"Start":"03:07.444 ","End":"03:12.590","Text":"and sine Theta in absolute value is less than or equal to 1."},{"Start":"03:12.590 ","End":"03:14.810","Text":"If I do cosine cubed Theta,"},{"Start":"03:14.810 ","End":"03:17.315","Text":"will still be less than 1 and sine Theta,"},{"Start":"03:17.315 ","End":"03:21.455","Text":"this whole thing is going to still be less than or equal to 1,"},{"Start":"03:21.455 ","End":"03:24.800","Text":"and so it\u0027s bounded."},{"Start":"03:24.800 ","End":"03:29.420","Text":"Only use the theorem that something it tends to 0,"},{"Start":"03:29.420 ","End":"03:33.100","Text":"that something bounded also goes to 0,"},{"Start":"03:33.100 ","End":"03:35.750","Text":"and that is the answer."},{"Start":"03:35.750 ","End":"03:40.200","Text":"I\u0027ll highlight it and declare that we are done."}],"ID":8912},{"Watched":false,"Name":"Exercise 4 part b","Duration":"7m 19s","ChapterTopicVideoID":8563,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.400","Text":"This time we have a limit as x and y both go to infinity,"},{"Start":"00:05.400 ","End":"00:06.420","Text":"and that makes a change."},{"Start":"00:06.420 ","End":"00:08.310","Text":"It\u0027s usually 0,0."},{"Start":"00:08.310 ","End":"00:13.110","Text":"In any event we can\u0027t just substitute as if infinity was a number,"},{"Start":"00:13.110 ","End":"00:14.535","Text":"which we sometimes can do,"},{"Start":"00:14.535 ","End":"00:19.050","Text":"but infinity minus infinity is 1 of those indeterminate forms,"},{"Start":"00:19.050 ","End":"00:22.395","Text":"or not defined, so we\u0027re stuck there."},{"Start":"00:22.395 ","End":"00:25.275","Text":"We\u0027ll have to use some technique."},{"Start":"00:25.275 ","End":"00:29.040","Text":"I already tried out proving and disproving."},{"Start":"00:29.040 ","End":"00:32.875","Text":"Turns out that the best thing to do is the polar substitution here."},{"Start":"00:32.875 ","End":"00:39.930","Text":"I\u0027ll remind you that the polar substitution says we let x equals r cosine theta,"},{"Start":"00:39.930 ","End":"00:43.885","Text":"and y equals r sine theta."},{"Start":"00:43.885 ","End":"00:47.150","Text":"We usually use it when x and y go to 0,0,"},{"Start":"00:47.150 ","End":"00:49.355","Text":"but it also worked for infinity."},{"Start":"00:49.355 ","End":"00:51.020","Text":"Only in this case,"},{"Start":"00:51.020 ","End":"00:54.960","Text":"r goes to infinity."},{"Start":"00:55.850 ","End":"01:00.149","Text":"Let\u0027s substitute, and we get the limit."},{"Start":"01:00.149 ","End":"01:08.465","Text":"As r goes to infinity of c,"},{"Start":"01:08.465 ","End":"01:18.660","Text":"x minus y is r cosine theta minus r sine theta,"},{"Start":"01:18.660 ","End":"01:27.180","Text":"over x squared is r squared, cosine squared theta,"},{"Start":"01:27.180 ","End":"01:37.080","Text":"here we have y times x is r squared cosine theta, sine theta."},{"Start":"01:38.690 ","End":"01:41.265","Text":"Here we have y^4,"},{"Start":"01:41.265 ","End":"01:48.825","Text":"which is r^4 sine^4 theta."},{"Start":"01:48.825 ","End":"01:52.695","Text":"We can simplify this a bit."},{"Start":"01:52.695 ","End":"01:56.490","Text":"We can take r out of the numerator,"},{"Start":"01:56.490 ","End":"01:59.120","Text":"and we can take r-squared out of the denominator,"},{"Start":"01:59.120 ","End":"02:05.044","Text":"so r over r squared will give us 1 over r. Let me just write the limit."},{"Start":"02:05.044 ","End":"02:06.710","Text":"It goes to infinity."},{"Start":"02:06.710 ","End":"02:09.090","Text":"That takes care of r here,"},{"Start":"02:09.090 ","End":"02:11.790","Text":"r-squared here, r-squared here, and r-squared here,"},{"Start":"02:11.790 ","End":"02:18.900","Text":"so all we\u0027re left with then is cosine theta minus sine theta."},{"Start":"02:23.470 ","End":"02:31.025","Text":"We have cosine squared theta plus"},{"Start":"02:31.025 ","End":"02:41.850","Text":"cosine theta sine theta plus r squared sine to the fourth theta."},{"Start":"02:42.250 ","End":"02:47.510","Text":"What I\u0027d like to say is that this expression is bounded."},{"Start":"02:47.510 ","End":"02:52.205","Text":"We\u0027ve had this before, because cosine and sine are between minus 1 and 1,"},{"Start":"02:52.205 ","End":"02:55.325","Text":"this kind of expression can\u0027t get very far."},{"Start":"02:55.325 ","End":"02:58.185","Text":"In this case, the most it could be 2,"},{"Start":"02:58.185 ","End":"03:00.450","Text":"the lowest it could be is minus 2."},{"Start":"03:00.450 ","End":"03:05.450","Text":"Similarly, because cosine and sine are between minus 1 and 1,"},{"Start":"03:05.450 ","End":"03:08.450","Text":"this can\u0027t get very far from 0."},{"Start":"03:08.450 ","End":"03:11.120","Text":"These are both bounded."},{"Start":"03:11.120 ","End":"03:16.000","Text":"This is bounded, and this is bounded."},{"Start":"03:16.000 ","End":"03:20.030","Text":"I was about to say that this goes to infinity,"},{"Start":"03:20.030 ","End":"03:21.830","Text":"but that\u0027s not quite right,"},{"Start":"03:21.830 ","End":"03:23.914","Text":"because there\u0027s an exception."},{"Start":"03:23.914 ","End":"03:30.650","Text":"Suppose that sine theta went to 0,"},{"Start":"03:30.650 ","End":"03:32.900","Text":"then we\u0027d have an infinity times 0,"},{"Start":"03:32.900 ","End":"03:34.550","Text":"which is not sure."},{"Start":"03:34.550 ","End":"03:42.365","Text":"I\u0027m going to take 2 cases that either sine infinity goes to 0, or does not."},{"Start":"03:42.365 ","End":"03:45.770","Text":"Let\u0027s say sine theta does not go to 0."},{"Start":"03:45.770 ","End":"03:47.690","Text":"If it doesn\u0027t go to 0,"},{"Start":"03:47.690 ","End":"03:48.770","Text":"then we have no problem."},{"Start":"03:48.770 ","End":"03:50.345","Text":"This goes to infinity,"},{"Start":"03:50.345 ","End":"03:52.055","Text":"and this is bounded,"},{"Start":"03:52.055 ","End":"03:57.130","Text":"so if we figure out what\u0027s going on here,"},{"Start":"03:57.130 ","End":"04:01.605","Text":"this is going to infinity,"},{"Start":"04:01.605 ","End":"04:06.150","Text":"this is bounded, this is also going to infinity,"},{"Start":"04:06.150 ","End":"04:12.160","Text":"we have 1 over infinity times bounded"},{"Start":"04:12.160 ","End":"04:21.135","Text":"over bounded, plus infinity."},{"Start":"04:21.135 ","End":"04:23.510","Text":"Bounded plus infinity is infinity,"},{"Start":"04:23.510 ","End":"04:28.280","Text":"bounded over infinity is 0,"},{"Start":"04:28.280 ","End":"04:30.379","Text":"1 over infinity is 0,"},{"Start":"04:30.379 ","End":"04:34.410","Text":"altogether we get 0."},{"Start":"04:34.410 ","End":"04:39.535","Text":"That takes care of the case where sine theta doesn\u0027t go to 0,"},{"Start":"04:39.535 ","End":"04:44.720","Text":"but now let\u0027s take care of that peculiar case where it does."},{"Start":"04:44.720 ","End":"04:49.295","Text":"If sine theta does go to 0,"},{"Start":"04:49.295 ","End":"04:54.545","Text":"then also sine squared theta goes to 0,"},{"Start":"04:54.545 ","End":"05:00.615","Text":"and we know that cosine squared theta then goes to 1."},{"Start":"05:00.615 ","End":"05:05.050","Text":"If cosine squared theta goes to 1,"},{"Start":"05:05.480 ","End":"05:07.890","Text":"now we have a variation."},{"Start":"05:07.890 ","End":"05:09.960","Text":"This goes to 1,"},{"Start":"05:09.960 ","End":"05:14.265","Text":"this also goes to 0 because sine theta goes to 0."},{"Start":"05:14.265 ","End":"05:15.840","Text":"This is bounded."},{"Start":"05:15.840 ","End":"05:19.910","Text":"This you can\u0027t tell what happens with infinity times 0,"},{"Start":"05:19.910 ","End":"05:24.800","Text":"but I certainly know that this is bigger or equal to 0,"},{"Start":"05:24.800 ","End":"05:27.710","Text":"because everything squared, or to the fourth."},{"Start":"05:27.710 ","End":"05:35.310","Text":"What it means is that the denominator is altogether bigger or equal to 1."},{"Start":"05:35.310 ","End":"05:41.000","Text":"Something bigger or equal to 1 times infinity is also infinity,"},{"Start":"05:41.000 ","End":"05:43.040","Text":"and I still get 1 over infinity,"},{"Start":"05:43.040 ","End":"05:45.575","Text":"or I get bounded over infinity,"},{"Start":"05:45.575 ","End":"05:49.430","Text":"so it still comes out to be 0."},{"Start":"05:49.430 ","End":"05:54.510","Text":"Let me just try that again. What we have is 1 over"},{"Start":"05:55.390 ","End":"06:05.220","Text":"infinity times bounded over,"},{"Start":"06:05.780 ","End":"06:16.810","Text":"we have 1 plus 0 plus bigger or equal to 0."},{"Start":"06:20.180 ","End":"06:24.200","Text":"We got bounded, this is infinity,"},{"Start":"06:24.200 ","End":"06:27.140","Text":"infinity over something bigger or equal to 1,"},{"Start":"06:27.140 ","End":"06:29.860","Text":"which is what we replaced this by,"},{"Start":"06:29.860 ","End":"06:33.090","Text":"altogether this was bigger or equal to 1."},{"Start":"06:33.090 ","End":"06:35.550","Text":"So bounded over infinity,"},{"Start":"06:35.550 ","End":"06:37.785","Text":"and so it\u0027s equal to 0."},{"Start":"06:37.785 ","End":"06:42.705","Text":"In this case, the limit, do I have room here?"},{"Start":"06:42.705 ","End":"06:45.230","Text":"No. I\u0027ll just put an arrow."},{"Start":"06:45.230 ","End":"06:49.460","Text":"This also goes to 0."},{"Start":"06:49.460 ","End":"06:56.109","Text":"We did separate 2 cases where sine theta goes to 0 or not,"},{"Start":"06:56.109 ","End":"06:58.985","Text":"and in this case,"},{"Start":"06:58.985 ","End":"07:04.650","Text":"we got that the limit was 0,"},{"Start":"07:04.650 ","End":"07:05.900","Text":"and in the other case,"},{"Start":"07:05.900 ","End":"07:10.025","Text":"we also got that the limit was 0, so either way,"},{"Start":"07:10.025 ","End":"07:12.770","Text":"the limit is 0,"},{"Start":"07:12.770 ","End":"07:16.215","Text":"and we are done."},{"Start":"07:16.215 ","End":"07:19.390","Text":"Yes, this was a bit of a tricky one."}],"ID":8913},{"Watched":false,"Name":"Exercise 4 part c","Duration":"3m 52s","ChapterTopicVideoID":8564,"CourseChapterTopicPlaylistID":4970,"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 compute"},{"Start":"00:02.430 ","End":"00:08.895","Text":"the following limit of function of 2 variables as x, y goes to 0, 0."},{"Start":"00:08.895 ","End":"00:11.910","Text":"The obvious thing we try, first of all,"},{"Start":"00:11.910 ","End":"00:13.945","Text":"is to just to substitute 0, 0,"},{"Start":"00:13.945 ","End":"00:16.710","Text":"but that doesn\u0027t work because we get"},{"Start":"00:16.710 ","End":"00:23.580","Text":"a 0 over 0 situation sine 0 over the square root of 0."},{"Start":"00:23.580 ","End":"00:27.555","Text":"We need to use some technique."},{"Start":"00:27.555 ","End":"00:29.490","Text":"We don\u0027t have that many techniques,"},{"Start":"00:29.490 ","End":"00:32.610","Text":"and just from trying some of them out,"},{"Start":"00:32.610 ","End":"00:35.550","Text":"it\u0027s polar substitution that\u0027s going to work."},{"Start":"00:35.550 ","End":"00:39.660","Text":"I want to remind you, polar substitution is when we let x equals"},{"Start":"00:39.660 ","End":"00:45.809","Text":"r cosine Theta and y equals r sine Theta."},{"Start":"00:45.809 ","End":"00:51.670","Text":"The other useful thing is that x squared plus y squared equals r squared."},{"Start":"00:51.670 ","End":"00:53.780","Text":"I\u0027m not going to derive it again."},{"Start":"00:53.780 ","End":"00:56.465","Text":"When x, y goes to 0, 0,"},{"Start":"00:56.465 ","End":"00:59.600","Text":"we replaced that by r goes to 0,"},{"Start":"00:59.600 ","End":"01:02.760","Text":"or strictly r goes to 0 from the right because of"},{"Start":"01:02.760 ","End":"01:08.254","Text":"the radius is taken as positive or non-negative."},{"Start":"01:08.254 ","End":"01:16.970","Text":"Substituting here, we get the limit as r goes to 0 of sine."},{"Start":"01:16.970 ","End":"01:25.505","Text":"Here we have r cosine Theta times r sine Theta, that\u0027s x, that\u0027s y."},{"Start":"01:25.505 ","End":"01:29.720","Text":"As I said, the square root of x squared plus y squared would be"},{"Start":"01:29.720 ","End":"01:32.990","Text":"the square root of r squared and"},{"Start":"01:32.990 ","End":"01:36.290","Text":"that\u0027s just r. Because if I take the square root of this,"},{"Start":"01:36.290 ","End":"01:39.090","Text":"just r, r is non-negative."},{"Start":"01:39.400 ","End":"01:45.155","Text":"What can we say? If we substitute r equals 0,"},{"Start":"01:45.155 ","End":"01:48.740","Text":"then again we get sine of 0,"},{"Start":"01:48.740 ","End":"01:50.900","Text":"which is 0 over 0."},{"Start":"01:50.900 ","End":"01:55.250","Text":"But here we have a function of 1 variable r so we can use L\u0027Hopital,"},{"Start":"01:55.250 ","End":"01:59.700","Text":"Theta\u0027s not dependent on r in the limit."},{"Start":"02:03.200 ","End":"02:12.690","Text":"Let me just say that this numerator I can write as r squared cosine Theta sine Theta."},{"Start":"02:13.210 ","End":"02:20.800","Text":"We get according to L\u0027Hopital writer\u0027s name,"},{"Start":"02:20.800 ","End":"02:26.069","Text":"L\u0027Hopital\u0027s rule that this is equal to the limit."},{"Start":"02:26.740 ","End":"02:30.800","Text":"If I differentiate the numerator,"},{"Start":"02:30.800 ","End":"02:33.650","Text":"I get cosine of the same thing,"},{"Start":"02:33.650 ","End":"02:41.670","Text":"r squared cosine Theta, sine Theta."},{"Start":"02:43.880 ","End":"02:48.325","Text":"But that\u0027s not all, I have to multiply by the inner derivative,"},{"Start":"02:48.325 ","End":"02:52.600","Text":"which is 2r, and also this constant,"},{"Start":"02:52.600 ","End":"02:54.355","Text":"constant as far as r goes."},{"Start":"02:54.355 ","End":"02:57.715","Text":"Cosine Theta, sine Theta,"},{"Start":"02:57.715 ","End":"02:59.920","Text":"and all this over,"},{"Start":"02:59.920 ","End":"03:02.575","Text":"well, derivative of this is just 1."},{"Start":"03:02.575 ","End":"03:05.530","Text":"We just have the numerator. Now, let\u0027s take a look."},{"Start":"03:05.530 ","End":"03:08.214","Text":"This is made up of 3 parts."},{"Start":"03:08.214 ","End":"03:12.735","Text":"This part here, let say this part here,"},{"Start":"03:12.735 ","End":"03:14.930","Text":"and this part here."},{"Start":"03:14.930 ","End":"03:18.085","Text":"Now, this part is bounded."},{"Start":"03:18.085 ","End":"03:19.570","Text":"Remember, when we have a cosine,"},{"Start":"03:19.570 ","End":"03:23.415","Text":"it\u0027s always between minus 1 and 1,"},{"Start":"03:23.415 ","End":"03:25.350","Text":"so it can\u0027t go very far."},{"Start":"03:25.350 ","End":"03:31.994","Text":"The 2r turns to 0 if I let r go to 0."},{"Start":"03:31.994 ","End":"03:34.400","Text":"This again is a cosine and the sine,"},{"Start":"03:34.400 ","End":"03:37.560","Text":"they\u0027re both bounded, so this is bounded."},{"Start":"03:37.880 ","End":"03:44.550","Text":"We know that bounded times 0 is 0,"},{"Start":"03:44.550 ","End":"03:48.225","Text":"so the answer is just 0."},{"Start":"03:48.225 ","End":"03:52.240","Text":"I\u0027ll highlight that and we\u0027re done."}],"ID":8914},{"Watched":false,"Name":"Exercise 4 part d","Duration":"2m 37s","ChapterTopicVideoID":8565,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.850","Text":"In this exercise, we have another limit of a function of 2 variables as x, y goes to 0,0."},{"Start":"00:08.850 ","End":"00:12.195","Text":"As usual, if we just try substituting,"},{"Start":"00:12.195 ","End":"00:13.830","Text":"we\u0027re going to get 0 over 0,"},{"Start":"00:13.830 ","End":"00:15.105","Text":"so that\u0027s no good."},{"Start":"00:15.105 ","End":"00:18.000","Text":"We don\u0027t have that many techniques at our disposal."},{"Start":"00:18.000 ","End":"00:21.960","Text":"Turns out that polar substitution is the one that\u0027s going to work"},{"Start":"00:21.960 ","End":"00:24.320","Text":"and just going to remind you that"},{"Start":"00:24.320 ","End":"00:32.445","Text":"we let x equals r cosine Theta and y equals r sine Theta."},{"Start":"00:32.445 ","End":"00:33.600","Text":"That\u0027s what\u0027s going to do it"},{"Start":"00:33.600 ","End":"00:38.459","Text":"and we also remember that x squared plus y squared equals r squared"},{"Start":"00:38.459 ","End":"00:49.010","Text":"and this limit is replaced by the limit as r goes to 0"},{"Start":"00:49.010 ","End":"00:53.345","Text":"or more precisely 0 from the positive direction."},{"Start":"00:53.345 ","End":"00:54.950","Text":"Let\u0027s do that."},{"Start":"00:54.950 ","End":"01:08.540","Text":"The limit as r goes to 0 of x to the 4 is r to the 4 cosine to the 4."},{"Start":"01:08.540 ","End":"01:13.100","Text":"Well, it doesn\u0027t matter because I can do it the other way around."},{"Start":"01:13.100 ","End":"01:17.285","Text":"Plus r to the 4 cosine to the 4 Theta."},{"Start":"01:17.285 ","End":"01:19.680","Text":"Let me just switch them around."},{"Start":"01:20.480 ","End":"01:26.680","Text":"Over x squared plus y squared is already written as r squared."},{"Start":"01:26.680 ","End":"01:31.580","Text":"Now, we can divide top and bottom by r squared"},{"Start":"01:31.580 ","End":"01:38.960","Text":"and we get the limit and we can also take out that r squared outside the brackets."},{"Start":"01:38.960 ","End":"01:43.250","Text":"Basically, we took r to the 4 out and divided it by r squared,"},{"Start":"01:43.250 ","End":"01:51.580","Text":"so we get r squared cosine to the 4 Theta plus sine to the 4 Theta."},{"Start":"01:51.710 ","End":"01:58.130","Text":"Now, we know we\u0027ve mentioned this a few times that cosine is bounded,"},{"Start":"01:58.130 ","End":"02:02.055","Text":"it\u0027s bounded in absolute value by 1,"},{"Start":"02:02.055 ","End":"02:03.630","Text":"it\u0027s between minus 1 and 1,"},{"Start":"02:03.630 ","End":"02:05.960","Text":"so to the 4th, it\u0027s going to be between 0 and 1."},{"Start":"02:05.960 ","End":"02:06.830","Text":"It\u0027s bounded."},{"Start":"02:06.830 ","End":"02:08.960","Text":"Similarly, sine is bounded,"},{"Start":"02:08.960 ","End":"02:11.360","Text":"so all this is bounded."},{"Start":"02:11.360 ","End":"02:15.410","Text":"It\u0027s actually bounded between, let\u0027s see 0 and 2."},{"Start":"02:15.410 ","End":"02:19.160","Text":"Each of them is positive and each of them is no bigger than 1."},{"Start":"02:19.160 ","End":"02:22.395","Text":"At most, it\u0027s between 0 and 2."},{"Start":"02:22.395 ","End":"02:25.430","Text":"This thing as r goes to 0, tends to 0"},{"Start":"02:25.430 ","End":"02:30.050","Text":"and there\u0027s a theorem that something goes to 0 times something that\u0027s bounded,"},{"Start":"02:30.050 ","End":"02:31.745","Text":"it also goes to 0."},{"Start":"02:31.745 ","End":"02:37.770","Text":"This limit is 0 and I\u0027ll highlight the answer and we\u0027re done."}],"ID":8915},{"Watched":false,"Name":"Exercise 4 part e","Duration":"4m 20s","ChapterTopicVideoID":8566,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"In this exercise, we have to compute the limit of a function"},{"Start":"00:05.130 ","End":"00:10.320","Text":"of 2 variables as x and y goes to 0,0."},{"Start":"00:10.320 ","End":"00:15.720","Text":"We just try to substitute 0,0 everywhere we\u0027ll get 0 over 0,"},{"Start":"00:15.720 ","End":"00:17.970","Text":"so that\u0027s not going to help."},{"Start":"00:17.970 ","End":"00:20.189","Text":"We have to try 1 of our techniques,"},{"Start":"00:20.189 ","End":"00:22.290","Text":"and after trial and error,"},{"Start":"00:22.290 ","End":"00:25.200","Text":"it\u0027s the polar substitution that\u0027s going to work."},{"Start":"00:25.200 ","End":"00:28.110","Text":"To remind you, polar substitution says,"},{"Start":"00:28.110 ","End":"00:31.905","Text":"we let x equal r cosine Theta,"},{"Start":"00:31.905 ","End":"00:36.810","Text":"and y equals r sine Theta."},{"Start":"00:36.810 ","End":"00:41.925","Text":"The limit as x, y goes to 0,0 is replaced by r goes to 0,"},{"Start":"00:41.925 ","End":"00:44.040","Text":"strictly, actually from above."},{"Start":"00:44.040 ","End":"00:47.310","Text":"The other useful thing is that x squared,"},{"Start":"00:47.310 ","End":"00:51.179","Text":"plus y squared equals r squared,"},{"Start":"00:51.179 ","End":"00:55.595","Text":"and that\u0027s because cosine squared plus sine squared is 1."},{"Start":"00:55.595 ","End":"00:58.970","Text":"If we put all that in our limit,"},{"Start":"00:58.970 ","End":"01:06.950","Text":"then we get the limit as r goes to 0 of 3x"},{"Start":"01:06.950 ","End":"01:16.500","Text":"squared is 3r squared cosine squared Theta."},{"Start":"01:17.930 ","End":"01:26.460","Text":"Then we have minus r squared cosine squared"},{"Start":"01:26.460 ","End":"01:34.580","Text":"Theta r squared sine squared Theta plus 3y squared;"},{"Start":"01:34.580 ","End":"01:38.975","Text":"is r squared sine squared Theta."},{"Start":"01:38.975 ","End":"01:43.040","Text":"All this over x squared plus y squared,"},{"Start":"01:43.040 ","End":"01:48.125","Text":"which we know is r squared. Let\u0027s see."},{"Start":"01:48.125 ","End":"01:51.590","Text":"Oh, and I forgot the squared here, sorry."},{"Start":"01:51.590 ","End":"01:58.050","Text":"Now, we can actually take r"},{"Start":"01:58.050 ","End":"02:04.929","Text":"squared outside the brackets above and below."},{"Start":"02:05.360 ","End":"02:12.525","Text":"What I\u0027m saying is that this r squared can factor into the r squared here."},{"Start":"02:12.525 ","End":"02:14.220","Text":"Let\u0027s see, 1 of these,"},{"Start":"02:14.220 ","End":"02:15.720","Text":"I\u0027ll take this 1,"},{"Start":"02:15.720 ","End":"02:24.395","Text":"and this 1, and so what we\u0027re left with is the limit as r goes to 0."},{"Start":"02:24.395 ","End":"02:26.300","Text":"Let\u0027s see, there\u0027s no denominator,"},{"Start":"02:26.300 ","End":"02:35.800","Text":"so it\u0027s 3 cosine squared Theta minus r squared,"},{"Start":"02:35.800 ","End":"02:41.419","Text":"cosine squared Theta sine squared"},{"Start":"02:41.419 ","End":"02:52.350","Text":"Theta plus 3 sine squared Theta."},{"Start":"02:52.430 ","End":"02:58.650","Text":"Again, I mentioned that sine squared of anything, it could be Theta,"},{"Start":"02:58.650 ","End":"03:02.400","Text":"could be Alpha but sine squared"},{"Start":"03:02.400 ","End":"03:07.470","Text":"Theta plus cosine squared Theta equals 1 for any angle Theta."},{"Start":"03:07.470 ","End":"03:11.310","Text":"Here, I have 3 cosine squared plus 3 sine squared."},{"Start":"03:11.310 ","End":"03:14.920","Text":"I can write this as the limit."},{"Start":"03:15.320 ","End":"03:18.345","Text":"This and this gives me 3,"},{"Start":"03:18.345 ","End":"03:27.580","Text":"minus r squared cosine squared Theta, sine squared Theta."},{"Start":"03:27.920 ","End":"03:34.700","Text":"Now, if I look at this,"},{"Start":"03:34.700 ","End":"03:39.240","Text":"this bit here is bounded."},{"Start":"03:39.290 ","End":"03:42.065","Text":"We\u0027ve mentioned this before."},{"Start":"03:42.065 ","End":"03:45.590","Text":"Cosine is between minus 1 and 1,"},{"Start":"03:45.590 ","End":"03:47.780","Text":"so cosine squared is between 0 and 1."},{"Start":"03:47.780 ","End":"03:49.955","Text":"Sine squared is between 0 and 1."},{"Start":"03:49.955 ","End":"03:52.340","Text":"Altogether, this can\u0027t go very far."},{"Start":"03:52.340 ","End":"03:53.885","Text":"It\u0027s between 0 and 1,"},{"Start":"03:53.885 ","End":"03:59.085","Text":"and r squared goes to 0,"},{"Start":"03:59.085 ","End":"04:00.630","Text":"as r goes to 0."},{"Start":"04:00.630 ","End":"04:03.555","Text":"For something that goes to 0 to something bounded,"},{"Start":"04:03.555 ","End":"04:07.569","Text":"this bit here goes to 0."},{"Start":"04:07.569 ","End":"04:10.415","Text":"I should have had brackets here by the way."},{"Start":"04:10.415 ","End":"04:14.209","Text":"All together 3 minus something that goes to 0,"},{"Start":"04:14.209 ","End":"04:16.820","Text":"so the limit will be 3,"},{"Start":"04:16.820 ","End":"04:18.350","Text":"which I shall highlight,"},{"Start":"04:18.350 ","End":"04:20.430","Text":"and we are done."}],"ID":8916},{"Watched":false,"Name":"Exercise 4 part f","Duration":"2m 41s","ChapterTopicVideoID":8567,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.179","Text":"In this exercise, we need to compute the following limit of expression in 2 variables,"},{"Start":"00:05.179 ","End":"00:12.300","Text":"x and y, and the thing to do is polar substitution."},{"Start":"00:12.300 ","End":"00:13.620","Text":"If you just try putting x,"},{"Start":"00:13.620 ","End":"00:15.060","Text":"y equals 0, 0,"},{"Start":"00:15.060 ","End":"00:19.290","Text":"you\u0027ll get sine of 0 over 0, that doesn\u0027t work."},{"Start":"00:19.290 ","End":"00:21.210","Text":"There\u0027s not that many techniques and turns"},{"Start":"00:21.210 ","End":"00:24.915","Text":"our polar substitution is what does give you a quick reminder."},{"Start":"00:24.915 ","End":"00:29.355","Text":"We let x equal r cosine Theta and y equals"},{"Start":"00:29.355 ","End":"00:37.115","Text":"r sine Theta and this limit is replaced by the limit r goes to 0 from the right."},{"Start":"00:37.115 ","End":"00:40.430","Text":"We also have the useful formula that x squared plus y"},{"Start":"00:40.430 ","End":"00:44.470","Text":"squared equals r squared. We\u0027ve seen this before."},{"Start":"00:44.470 ","End":"00:51.170","Text":"This limit becomes the limit as r goes to 0."},{"Start":"00:51.170 ","End":"00:54.710","Text":"Now, if x squared plus y squared equals r squared,"},{"Start":"00:54.710 ","End":"01:02.135","Text":"this is the square root of r squared just comes out to be r. We have sine of"},{"Start":"01:02.135 ","End":"01:13.210","Text":"r over the cube root of r squared is r^2/3."},{"Start":"01:14.600 ","End":"01:18.565","Text":"Now, there\u0027s a couple of ways I could go from here."},{"Start":"01:18.565 ","End":"01:22.350","Text":"Just substituting r equals 0 won\u0027t help because we\u0027ve got"},{"Start":"01:22.350 ","End":"01:26.309","Text":"0 over 0 same as we did originally,"},{"Start":"01:26.309 ","End":"01:28.565","Text":"and we could use L\u0027Hopital\u0027s rule."},{"Start":"01:28.565 ","End":"01:30.770","Text":"But here\u0027s another way we could go."},{"Start":"01:30.770 ","End":"01:35.420","Text":"We generally know that the famous limit that whenever something goes to 0,"},{"Start":"01:35.420 ","End":"01:40.970","Text":"we\u0027ll call it Alpha of sine of that thing over that same thing it\u0027s equal to 1."},{"Start":"01:40.970 ","End":"01:43.970","Text":"I would like to use this method instead of L\u0027Hopital."},{"Start":"01:43.970 ","End":"01:49.870","Text":"All I have to do is multiply top and bottom by r^1/3."},{"Start":"01:49.870 ","End":"01:55.070","Text":"If I put r^1/3 here and r^1/3 here,"},{"Start":"01:55.070 ","End":"02:02.030","Text":"then I haven\u0027t changed anything but r^2/3 times r^1/3 is r. This gives us"},{"Start":"02:02.030 ","End":"02:10.740","Text":"the limit as r goes to 0 of sine r over r from these 2,"},{"Start":"02:10.740 ","End":"02:17.600","Text":"times r^1/3 which I could write again as the cube root of r, doesn\u0027t really matter."},{"Start":"02:17.600 ","End":"02:19.450","Text":"I just like this better."},{"Start":"02:19.450 ","End":"02:29.120","Text":"Now, this tends to 1 because of this famous limit here and when r goes to 0,"},{"Start":"02:29.120 ","End":"02:32.410","Text":"the cube root of r also goes to 0."},{"Start":"02:32.410 ","End":"02:35.465","Text":"This whole limit is 1 times 0,"},{"Start":"02:35.465 ","End":"02:41.040","Text":"which is 0 and I\u0027ll just highlight it and that\u0027s the answer. We\u0027re done."}],"ID":8917},{"Watched":false,"Name":"Exercise 4 part g","Duration":"5m 7s","ChapterTopicVideoID":8568,"CourseChapterTopicPlaylistID":4970,"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, you have to compute this limit of 2 variables."},{"Start":"00:04.380 ","End":"00:08.565","Text":"If we just let x and y be 0 and 0"},{"Start":"00:08.565 ","End":"00:13.365","Text":"we\u0027ll get 0 times natural log of 0."},{"Start":"00:13.365 ","End":"00:19.645","Text":"Actually, 0 plus, because these are not negative."},{"Start":"00:19.645 ","End":"00:25.430","Text":"Natural log of 0 plus is minus infinity."},{"Start":"00:25.430 ","End":"00:28.675","Text":"We get 0 times minus infinity."},{"Start":"00:28.675 ","End":"00:32.165","Text":"That\u0027s an undefined indeterminate form."},{"Start":"00:32.165 ","End":"00:34.355","Text":"We can\u0027t just substitute."},{"Start":"00:34.355 ","End":"00:37.670","Text":"We need to use 1 of our techniques and turns out"},{"Start":"00:37.670 ","End":"00:40.610","Text":"that polar substitution is what\u0027s going to do it."},{"Start":"00:40.610 ","End":"00:50.200","Text":"To remind you, we let x equal r cosine Theta and y equals r sine Theta."},{"Start":"00:50.200 ","End":"00:52.160","Text":"If we make this substitution,"},{"Start":"00:52.160 ","End":"00:54.979","Text":"the limit becomes r goes to 0,"},{"Start":"00:54.979 ","End":"00:58.910","Text":"but from above through positives."},{"Start":"00:58.910 ","End":"01:04.970","Text":"There\u0027s also another handy formula because sine squared plus cosine squared is 1."},{"Start":"01:04.970 ","End":"01:09.095","Text":"That x squared plus y squared equals r squared. This will be useful."},{"Start":"01:09.095 ","End":"01:10.700","Text":"Back to the limit."},{"Start":"01:10.700 ","End":"01:12.985","Text":"This becomes the limit."},{"Start":"01:12.985 ","End":"01:15.885","Text":"As r goes to 0,"},{"Start":"01:15.885 ","End":"01:27.120","Text":"y is r sine Theta and this becomes the natural log of r squared."},{"Start":"01:28.250 ","End":"01:32.820","Text":"Now natural log of r squared,"},{"Start":"01:32.820 ","End":"01:35.770","Text":"just in case you forgotten the rules of the logarithm,"},{"Start":"01:35.770 ","End":"01:43.760","Text":"natural log in general of a to the power of b is b times natural log of a,"},{"Start":"01:43.760 ","End":"01:46.420","Text":"actually worked for any log there,"},{"Start":"01:46.420 ","End":"01:49.300","Text":"put the general logarithm to any base,"},{"Start":"01:49.300 ","End":"01:50.920","Text":"in particular to base e,"},{"Start":"01:50.920 ","End":"01:53.065","Text":"which is a natural logarithm."},{"Start":"01:53.065 ","End":"01:55.684","Text":"What we get is,"},{"Start":"01:55.684 ","End":"01:58.575","Text":"this is 2 natural log of r,"},{"Start":"01:58.575 ","End":"02:00.630","Text":"actually the 2 I could put in front of the limit."},{"Start":"02:00.630 ","End":"02:05.575","Text":"We have 2 times the limit as r goes to 0"},{"Start":"02:05.575 ","End":"02:12.624","Text":"of sine Theta times natural log,"},{"Start":"02:12.624 ","End":"02:16.105","Text":"sorry, and there\u0027s an r. Yeah,"},{"Start":"02:16.105 ","End":"02:21.490","Text":"I just prefer to put the sine Theta in front than the r natural log"},{"Start":"02:21.490 ","End":"02:27.389","Text":"of r. Now this thing here is bounded,"},{"Start":"02:27.389 ","End":"02:28.500","Text":"we\u0027ve seen this before,"},{"Start":"02:28.500 ","End":"02:31.455","Text":"it\u0027s between minus 1 and 1."},{"Start":"02:31.455 ","End":"02:42.310","Text":"I\u0027m claiming that this goes to 0. I\u0027ll show you why in a minute."},{"Start":"02:42.310 ","End":"02:46.550","Text":"I\u0027ll put a little asterisk to remind me to show you why this is so."},{"Start":"02:46.730 ","End":"02:51.265","Text":"As we know, something bounded time something that goes to 0,"},{"Start":"02:51.265 ","End":"02:54.010","Text":"goes to 0, and of course,"},{"Start":"02:54.010 ","End":"02:57.235","Text":"times 2 is still 0. This is 0."},{"Start":"02:57.235 ","End":"03:00.290","Text":"Now I have to show you the asterisk."},{"Start":"03:02.150 ","End":"03:07.590","Text":"The limit as r goes to 0,"},{"Start":"03:07.590 ","End":"03:11.330","Text":"it\u0027s good that r goes to 0 from above because"},{"Start":"03:11.330 ","End":"03:14.780","Text":"the natural log is only defined for positives, anyway,"},{"Start":"03:14.780 ","End":"03:18.445","Text":"of r natural log of r,"},{"Start":"03:18.445 ","End":"03:20.480","Text":"can\u0027t substitute like I said,"},{"Start":"03:20.480 ","End":"03:22.820","Text":"this gives 0 times minus infinity."},{"Start":"03:22.820 ","End":"03:27.980","Text":"But what we can do is the 0 times infinity case as a standard trick."},{"Start":"03:27.980 ","End":"03:31.475","Text":"The standard trick is to,"},{"Start":"03:31.475 ","End":"03:34.310","Text":"instead of putting r in the numerator,"},{"Start":"03:34.310 ","End":"03:39.620","Text":"I can put it as 1 over r in the denominator."},{"Start":"03:39.620 ","End":"03:43.815","Text":"I mean, multiplying by r or dividing by"},{"Start":"03:43.815 ","End":"03:46.340","Text":"1 over r is the same thing because"},{"Start":"03:46.340 ","End":"03:49.475","Text":"dividing by a fraction is like multiplying by the reciprocal."},{"Start":"03:49.475 ","End":"03:55.385","Text":"Now this is an infinity over infinity situation."},{"Start":"03:55.385 ","End":"03:58.400","Text":"Well, actually it\u0027s minus infinity over infinity."},{"Start":"03:58.400 ","End":"04:01.355","Text":"When r goes to 0 from positive,"},{"Start":"04:01.355 ","End":"04:03.995","Text":"this goes to infinity, that goes to minus infinity."},{"Start":"04:03.995 ","End":"04:09.510","Text":"We can use the L\u0027Hopital rule here,"},{"Start":"04:09.510 ","End":"04:11.000","Text":"I\u0027ll write his name,"},{"Start":"04:11.000 ","End":"04:13.429","Text":"and he said that in such situations,"},{"Start":"04:13.429 ","End":"04:15.769","Text":"0 over 0 or infinity over infinity,"},{"Start":"04:15.769 ","End":"04:19.115","Text":"we can differentiate the top and differentiate the bottom"},{"Start":"04:19.115 ","End":"04:24.590","Text":"and get another limit that will have the same result."},{"Start":"04:24.590 ","End":"04:26.900","Text":"We have the limit as r goes to 0,"},{"Start":"04:26.900 ","End":"04:30.500","Text":"natural log gives us 1 over r."},{"Start":"04:31.530 ","End":"04:36.690","Text":"1 over r gives us minus 1 over r squared, it\u0027s well-known."},{"Start":"04:36.690 ","End":"04:40.350","Text":"This becomes, let\u0027s see,"},{"Start":"04:40.350 ","End":"04:44.530","Text":"this would be the limit."},{"Start":"04:44.530 ","End":"04:46.850","Text":"I\u0027m just going to do some algebra here."},{"Start":"04:46.850 ","End":"04:47.930","Text":"If I divide it out,"},{"Start":"04:47.930 ","End":"04:49.595","Text":"r squared goes to the top."},{"Start":"04:49.595 ","End":"04:53.840","Text":"It\u0027s minus r squared over r,"},{"Start":"04:53.840 ","End":"04:57.920","Text":"it\u0027s minus r basically and r goes to 0,"},{"Start":"04:57.920 ","End":"05:01.850","Text":"then minus r also goes to 0."},{"Start":"05:01.850 ","End":"05:03.410","Text":"That\u0027s our answer."},{"Start":"05:03.410 ","End":"05:08.250","Text":"Like we said earlier on, and we\u0027re done."}],"ID":8918},{"Watched":false,"Name":"Exercise 4 part h","Duration":"5m 17s","ChapterTopicVideoID":8569,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.920","Text":"What we have this time is the limit in 3 variables; x,"},{"Start":"00:04.920 ","End":"00:08.505","Text":"y, and z, and they all go to 0,"},{"Start":"00:08.505 ","End":"00:15.540","Text":"and this limit also doesn\u0027t just work by substitution because if I let x, y,"},{"Start":"00:15.540 ","End":"00:18.810","Text":"and z all be 0, then I get 0 on the top,"},{"Start":"00:18.810 ","End":"00:19.950","Text":"0 on the bottom,"},{"Start":"00:19.950 ","End":"00:23.985","Text":"I get a 0 over 0 situation, and that\u0027s no good."},{"Start":"00:23.985 ","End":"00:32.265","Text":"Now, 1 of the techniques that we have for 3 variables in 3 dimensions,"},{"Start":"00:32.265 ","End":"00:34.800","Text":"and it may not have been covered in the tutorial,"},{"Start":"00:34.800 ","End":"00:36.700","Text":"but I\u0027ll present it here,"},{"Start":"00:36.700 ","End":"00:43.865","Text":"is the transformation or substitution to spherical coordinates."},{"Start":"00:43.865 ","End":"00:46.820","Text":"Just like we had polar coordinates in 2D."},{"Start":"00:46.820 ","End":"00:50.390","Text":"In 3D, besides the Cartesian,"},{"Start":"00:50.390 ","End":"00:52.770","Text":"we have cylindrical or spherical,"},{"Start":"00:52.770 ","End":"00:57.065","Text":"and it turns out that the spherical coordinates,"},{"Start":"00:57.065 ","End":"00:59.340","Text":"what\u0027s going to help us here."},{"Start":"00:59.340 ","End":"01:02.885","Text":"Spherical substitution, and there\u0027s a standard formula,"},{"Start":"01:02.885 ","End":"01:05.930","Text":"and I\u0027m assuming that you\u0027ve seen spherical coordinates."},{"Start":"01:05.930 ","End":"01:08.090","Text":"I\u0027ll just give the formulas."},{"Start":"01:08.090 ","End":"01:11.300","Text":"In spherical, we have 3 variables; Rho,"},{"Start":"01:11.300 ","End":"01:15.589","Text":"sometimes called r, Theta, and Phi."},{"Start":"01:15.589 ","End":"01:18.364","Text":"This is like distance to the origin."},{"Start":"01:18.364 ","End":"01:23.720","Text":"It\u0027s a kind of longitude and this is a kind of latitude."},{"Start":"01:23.720 ","End":"01:28.700","Text":"The transformation rules are that x"},{"Start":"01:28.700 ","End":"01:36.750","Text":"equals Rho cosine Theta sine of Phi,"},{"Start":"01:36.750 ","End":"01:43.590","Text":"y equals Rho sine Theta also sine Phi,"},{"Start":"01:43.590 ","End":"01:49.620","Text":"and z equals Rho cosine of Phi."},{"Start":"01:49.620 ","End":"01:53.070","Text":"When we have x, y, z going to 0,"},{"Start":"01:53.070 ","End":"01:58.535","Text":"that becomes Rho goes to 0 from above."},{"Start":"01:58.535 ","End":"02:01.100","Text":"There is also a useful formula."},{"Start":"02:01.100 ","End":"02:06.290","Text":"If you compute x squared plus y squared plus z squared,"},{"Start":"02:06.290 ","End":"02:13.294","Text":"then it comes out to be exactly Rho squared."},{"Start":"02:13.294 ","End":"02:15.290","Text":"Some people use r instead of Rho,"},{"Start":"02:15.290 ","End":"02:21.485","Text":"but I don\u0027t want to confuse it with cylindrical where we do use r. Now back to the limit."},{"Start":"02:21.485 ","End":"02:27.760","Text":"We have limit as Rho goes to 0."},{"Start":"02:27.760 ","End":"02:30.620","Text":"Let\u0027s see now, we need to substitute,"},{"Start":"02:30.620 ","End":"02:32.675","Text":"well, the denominator is the easiest. You know what?"},{"Start":"02:32.675 ","End":"02:35.020","Text":"I\u0027ll start with the denominator."},{"Start":"02:35.020 ","End":"02:38.160","Text":"The denominator is just Rho squared."},{"Start":"02:38.160 ","End":"02:39.210","Text":"Now, let\u0027s see the numerator,"},{"Start":"02:39.210 ","End":"02:41.415","Text":"x cubed is"},{"Start":"02:41.415 ","End":"02:51.990","Text":"Rho cubed cosine cubed Theta sine cubed Phi."},{"Start":"02:51.990 ","End":"02:56.735","Text":"Y cubed is Rho cubed"},{"Start":"02:56.735 ","End":"03:06.935","Text":"sine cubed Theta sine cubed Phi."},{"Start":"03:06.935 ","End":"03:16.440","Text":"Z cubed is Rho cubed cosine cubed Phi."},{"Start":"03:18.370 ","End":"03:24.270","Text":"This is a bit longer. Now, we can"},{"Start":"03:24.270 ","End":"03:29.985","Text":"take Rho cubed out of the brackets here."},{"Start":"03:29.985 ","End":"03:33.900","Text":"Rho cubed over Rho squared will just give us Rho."},{"Start":"03:33.900 ","End":"03:38.775","Text":"What we get is the limit as Rho goes to 0 plus."},{"Start":"03:38.775 ","End":"03:44.735","Text":"As I said, Rho cubed, Rho cubed comes out as Rho cubed over Rho squared is just Rho."},{"Start":"03:44.735 ","End":"03:47.895","Text":"That was to indicate that that Rho squared goes in,"},{"Start":"03:47.895 ","End":"03:50.400","Text":"cancels with the 3, so to speak."},{"Start":"03:50.400 ","End":"03:53.200","Text":"Then we just have Rho."},{"Start":"03:55.940 ","End":"04:04.200","Text":"Well, let\u0027s see. Yeah, just copy the rest of it."},{"Start":"04:04.200 ","End":"04:07.320","Text":"It\u0027s a bit of a bore to write out without trying to do it quickly."},{"Start":"04:07.320 ","End":"04:13.430","Text":"Cosine cubed Theta sine cubed Phi is actually not that important as you\u0027ll see why,"},{"Start":"04:13.430 ","End":"04:14.480","Text":"so I\u0027m just rushing through"},{"Start":"04:14.480 ","End":"04:22.235","Text":"that sine cubed Theta sine cubed Phi plus cosine cubed Phi, closed brackets."},{"Start":"04:22.235 ","End":"04:24.050","Text":"The point I\u0027m trying to make is,"},{"Start":"04:24.050 ","End":"04:25.320","Text":"and the reason this is not important,"},{"Start":"04:25.320 ","End":"04:28.625","Text":"is that cosine and sine of anything,"},{"Start":"04:28.625 ","End":"04:31.520","Text":"cosine of whatever, it doesn\u0027t matter,"},{"Start":"04:31.520 ","End":"04:33.995","Text":"Theta or Phi or anything,"},{"Start":"04:33.995 ","End":"04:35.840","Text":"is bounded by 1,"},{"Start":"04:35.840 ","End":"04:38.600","Text":"it\u0027s between plus and minus 1."},{"Start":"04:38.600 ","End":"04:43.310","Text":"The same thing, the sine of whatever is also less than or equal to 1."},{"Start":"04:43.310 ","End":"04:45.860","Text":"Cosine and sine are bounded."},{"Start":"04:45.860 ","End":"04:50.605","Text":"At most, this expression could be 3."},{"Start":"04:50.605 ","End":"04:52.980","Text":"I mean, things that are less than or equal to 1"},{"Start":"04:52.980 ","End":"04:55.865","Text":"to any power is still less than or equal to 1."},{"Start":"04:55.865 ","End":"04:58.295","Text":"This whole thing is bounded."},{"Start":"04:58.295 ","End":"05:06.870","Text":"What I have is that this goes to 0 and this bit here is bounded,"},{"Start":"05:06.870 ","End":"05:12.230","Text":"and we already know that 0 times bounded is 0,"},{"Start":"05:12.230 ","End":"05:14.615","Text":"so the answer is 0,"},{"Start":"05:14.615 ","End":"05:16.115","Text":"which I shall highlight,"},{"Start":"05:16.115 ","End":"05:18.270","Text":"and that\u0027s the end."}],"ID":8919},{"Watched":false,"Name":"Exercise 5 part a","Duration":"4m 36s","ChapterTopicVideoID":8570,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.005","Text":"In this exercise, we\u0027re given a function as defined below in piece-wise manner."},{"Start":"00:07.005 ","End":"00:11.775","Text":"We have to check if it\u0027s continuous at the 0, 0."},{"Start":"00:11.775 ","End":"00:15.105","Text":"If so, all is well, and if not,"},{"Start":"00:15.105 ","End":"00:20.325","Text":"the question is, is the discontinuity removable?"},{"Start":"00:20.325 ","End":"00:24.135","Text":"Let\u0027s start and check continuity."},{"Start":"00:24.135 ","End":"00:31.260","Text":"Now, remember continuity means that the limit at the point equals the value at the point."},{"Start":"00:31.260 ","End":"00:35.835","Text":"I could even summarize it that limit at the point"},{"Start":"00:35.835 ","End":"00:40.395","Text":"equals the value at the point and that\u0027s what continuity means."},{"Start":"00:40.395 ","End":"00:44.620","Text":"Continuity in short. Let\u0027s, first of all,"},{"Start":"00:44.620 ","End":"00:47.105","Text":"do the limit at the point,"},{"Start":"00:47.105 ","End":"00:50.060","Text":"which is the limit as x,"},{"Start":"00:50.060 ","End":"00:54.620","Text":"y goes to 0,"},{"Start":"00:54.620 ","End":"00:59.975","Text":"0 of f of x and y."},{"Start":"00:59.975 ","End":"01:02.960","Text":"Now this equals because x,"},{"Start":"01:02.960 ","End":"01:05.195","Text":"y goes to 0, 0,"},{"Start":"01:05.195 ","End":"01:07.715","Text":"it\u0027s not equal to 0, 0,"},{"Start":"01:07.715 ","End":"01:12.410","Text":"and therefore, or in this case here so I\u0027m going to copy this bit."},{"Start":"01:12.410 ","End":"01:14.280","Text":"We get the limit,"},{"Start":"01:14.280 ","End":"01:17.915","Text":"same thing, x, y goes to 0,"},{"Start":"01:17.915 ","End":"01:23.030","Text":"0 of sine of x squared plus y"},{"Start":"01:23.030 ","End":"01:28.945","Text":"squared over x squared plus y squared."},{"Start":"01:28.945 ","End":"01:35.345","Text":"Now, we\u0027re going to use the technique of substitution to compute this limit."},{"Start":"01:35.345 ","End":"01:38.905","Text":"What we\u0027re going to do is let"},{"Start":"01:38.905 ","End":"01:42.150","Text":"t equal x squared plus y"},{"Start":"01:42.150 ","End":"01:45.380","Text":"squared because I see that everywhere and I get the same expression,"},{"Start":"01:45.380 ","End":"01:47.705","Text":"x squared plus y squared so we substitute it."},{"Start":"01:47.705 ","End":"01:52.205","Text":"Now when x, y goes to 0,"},{"Start":"01:52.205 ","End":"01:57.020","Text":"0, then t goes to 0 squared plus 0 squared, which is 0."},{"Start":"01:57.020 ","End":"02:00.140","Text":"Then t also goes to 0."},{"Start":"02:00.140 ","End":"02:02.840","Text":"Matter of fact, it goes to 0 from above,"},{"Start":"02:02.840 ","End":"02:04.190","Text":"but that doesn\u0027t matter."},{"Start":"02:04.190 ","End":"02:09.320","Text":"After we substitute, we get the limit as t goes to"},{"Start":"02:09.320 ","End":"02:18.110","Text":"0 of sine of t over t. Now,"},{"Start":"02:18.110 ","End":"02:20.225","Text":"this is a famous limit,"},{"Start":"02:20.225 ","End":"02:22.460","Text":"sine of something go over the same something."},{"Start":"02:22.460 ","End":"02:27.695","Text":"When it goes to 0, this is equal to 1 and that\u0027s"},{"Start":"02:27.695 ","End":"02:33.955","Text":"the limit part and I\u0027ll highlight it."},{"Start":"02:33.955 ","End":"02:38.000","Text":"Just in case some of you are not sure about this limit,"},{"Start":"02:38.000 ","End":"02:41.330","Text":"I\u0027ll show you how it\u0027s done and actually,"},{"Start":"02:41.330 ","End":"02:42.860","Text":"it\u0027s very quick to compute."},{"Start":"02:42.860 ","End":"02:44.750","Text":"We can use the L\u0027Hopitals theorem."},{"Start":"02:44.750 ","End":"02:49.280","Text":"Remember L\u0027Hopital who said that if you have a 0 over 0 situation,"},{"Start":"02:49.280 ","End":"02:53.165","Text":"then you can differentiate the top and bottom separately."},{"Start":"02:53.165 ","End":"02:56.480","Text":"Derivative of sine t is cosine t,"},{"Start":"02:56.480 ","End":"03:00.210","Text":"derivative of t is 1 and when t goes to 0,"},{"Start":"03:00.210 ","End":"03:01.920","Text":"we just get cosine 0,"},{"Start":"03:01.920 ","End":"03:05.415","Text":"which is equal to 1 so that confirms this."},{"Start":"03:05.415 ","End":"03:09.740","Text":"That was the limit part now we want to go for the value part."},{"Start":"03:09.740 ","End":"03:13.775","Text":"That just means that I take my function,"},{"Start":"03:13.775 ","End":"03:16.220","Text":"f at the point 0, 0."},{"Start":"03:16.220 ","End":"03:17.690","Text":"Now if x, y is 0, 0,"},{"Start":"03:17.690 ","End":"03:19.880","Text":"I\u0027m reading off this last line,"},{"Start":"03:19.880 ","End":"03:24.355","Text":"I get f of 0, 0 and this is equal to 2."},{"Start":"03:24.355 ","End":"03:27.860","Text":"Now, this 2 and this 1 are not equal."},{"Start":"03:27.860 ","End":"03:36.150","Text":"I said we have to have that limit equals value and this is not the case here."},{"Start":"03:36.350 ","End":"03:44.070","Text":"I can write the answer to the first question as not continuous,"},{"Start":"03:44.070 ","End":"03:46.610","Text":"but in the case of not continuous,"},{"Start":"03:46.610 ","End":"03:48.650","Text":"we\u0027re asked another question,"},{"Start":"03:48.650 ","End":"03:51.905","Text":"can we remove the discontinuity?"},{"Start":"03:51.905 ","End":"03:56.480","Text":"In other words, can we redefine the value at the point to make it continuous?"},{"Start":"03:56.480 ","End":"03:57.815","Text":"The answer, of course, is yes."},{"Start":"03:57.815 ","End":"03:59.720","Text":"The only problem is that 1 equals 2,"},{"Start":"03:59.720 ","End":"04:00.950","Text":"I can\u0027t change the limit,"},{"Start":"04:00.950 ","End":"04:02.900","Text":"but I can change the value."},{"Start":"04:02.900 ","End":"04:05.990","Text":"If we let, instead of this,"},{"Start":"04:05.990 ","End":"04:08.600","Text":"if we defined f of 0,"},{"Start":"04:08.600 ","End":"04:11.615","Text":"0 redefined to equal 1,"},{"Start":"04:11.615 ","End":"04:13.264","Text":"then we\u0027d be okay."},{"Start":"04:13.264 ","End":"04:18.630","Text":"Let me just copy this function."},{"Start":"04:19.090 ","End":"04:21.860","Text":"Here it is. I want to show you precisely what I mean."},{"Start":"04:21.860 ","End":"04:24.485","Text":"Instead of the 2, I\u0027m going to erase the 2,"},{"Start":"04:24.485 ","End":"04:31.335","Text":"there it\u0027s gone and instead of the 2 to write a 1 so we have a 1 here."},{"Start":"04:31.335 ","End":"04:37.110","Text":"Now, this function is continuous and we\u0027re done."}],"ID":8920},{"Watched":false,"Name":"Exercise 5 part b","Duration":"4m 57s","ChapterTopicVideoID":8571,"CourseChapterTopicPlaylistID":4970,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.400","Text":"In this exercise we\u0027re given a function defined here piece-wise."},{"Start":"00:05.400 ","End":"00:12.060","Text":"We have to check if it\u0027s continuous at the point 0, 0."},{"Start":"00:12.060 ","End":"00:19.470","Text":"If it isn\u0027t then we need to find out if this discontinuity is removable."},{"Start":"00:19.470 ","End":"00:22.825","Text":"Well, let\u0027s see. Let\u0027s first of all check continuity."},{"Start":"00:22.825 ","End":"00:29.040","Text":"Now continuity, just to say what it means."},{"Start":"00:29.040 ","End":"00:31.340","Text":"Continuity means it that"},{"Start":"00:31.340 ","End":"00:35.450","Text":"the limit of the function at the point equals the value at the point."},{"Start":"00:35.450 ","End":"00:38.720","Text":"I\u0027ll just write the shorthand that limit equals value."},{"Start":"00:38.720 ","End":"00:40.430","Text":"It\u0027s just like a mnemonic."},{"Start":"00:40.430 ","End":"00:42.470","Text":"Let\u0027s go for the limit part first,"},{"Start":"00:42.470 ","End":"00:45.635","Text":"what we want is the limit as x,"},{"Start":"00:45.635 ","End":"00:49.429","Text":"y goes to 0,"},{"Start":"00:49.429 ","End":"00:51.990","Text":"0, that\u0027s the point."},{"Start":"00:53.120 ","End":"00:55.590","Text":"When we\u0027re going to 0, 0,"},{"Start":"00:55.590 ","End":"00:57.150","Text":"we\u0027re not equaling 0,"},{"Start":"00:57.150 ","End":"01:01.530","Text":"0 so we look at this definition from here."},{"Start":"01:01.640 ","End":"01:04.140","Text":"First of all, write it as f of x,"},{"Start":"01:04.140 ","End":"01:05.630","Text":"y and then we\u0027ll write,"},{"Start":"01:05.630 ","End":"01:08.310","Text":"which means that this is equal to the limit as x,"},{"Start":"01:08.310 ","End":"01:09.860","Text":"y goes to 0,"},{"Start":"01:09.860 ","End":"01:11.390","Text":"0 of this part,"},{"Start":"01:11.390 ","End":"01:18.900","Text":"x cubed plus y cubed over x squared plus y squared."},{"Start":"01:18.900 ","End":"01:21.050","Text":"We\u0027ve seen similar limits before."},{"Start":"01:21.050 ","End":"01:26.615","Text":"You can probably guess that the polar substitution is what we need here."},{"Start":"01:26.615 ","End":"01:28.385","Text":"I\u0027ll remind you what that is."},{"Start":"01:28.385 ","End":"01:32.240","Text":"We let x equal r cosine Theta,"},{"Start":"01:32.240 ","End":"01:37.120","Text":"y equals r sine Theta."},{"Start":"01:37.120 ","End":"01:40.385","Text":"When x, y goes to 0, 0,"},{"Start":"01:40.385 ","End":"01:46.290","Text":"the limit is replaced by r goes to 0, but from above."},{"Start":"01:47.120 ","End":"01:53.140","Text":"One of the other things is that r squared equals x squared plus y squared,"},{"Start":"01:53.140 ","End":"01:54.980","Text":"usually write it the other way around,"},{"Start":"01:54.980 ","End":"02:00.510","Text":"and then r is the square root of x squared plus y squared, which is positive."},{"Start":"02:00.550 ","End":"02:05.720","Text":"We replace this limit by the limit as r goes to 0,"},{"Start":"02:05.720 ","End":"02:08.870","Text":"but others say technically it\u0027s 0 from above."},{"Start":"02:08.870 ","End":"02:11.125","Text":"Let\u0027s see what we have here."},{"Start":"02:11.125 ","End":"02:12.590","Text":"The denominator is easier,"},{"Start":"02:12.590 ","End":"02:15.200","Text":"already spelled it out for you, that\u0027s r squared."},{"Start":"02:15.200 ","End":"02:16.865","Text":"What about the numerator?"},{"Start":"02:16.865 ","End":"02:18.790","Text":"Well, just x and y separately."},{"Start":"02:18.790 ","End":"02:25.510","Text":"Here we have r cubed cosine cubed Theta and here,"},{"Start":"02:25.510 ","End":"02:30.925","Text":"r cubed sine cubed Theta."},{"Start":"02:30.925 ","End":"02:33.260","Text":"Of course, when I take something to the power of 3,"},{"Start":"02:33.260 ","End":"02:39.380","Text":"I take this bit separately to the 1/3 and this bit separately to the 1/3, simple algebra."},{"Start":"02:39.470 ","End":"02:44.810","Text":"Now we can cancel top and bottom by r squared."},{"Start":"02:44.810 ","End":"02:48.200","Text":"It\u0027s like I took r cubed out and then r cubed over r squared cancels."},{"Start":"02:48.200 ","End":"02:54.440","Text":"Let me just write this symbolically as r squared cancels with r squared here,"},{"Start":"02:54.440 ","End":"02:56.690","Text":"so we\u0027re just left with r. I just put a line through the"},{"Start":"02:56.690 ","End":"03:00.215","Text":"3 to show that we have r and r here."},{"Start":"03:00.215 ","End":"03:02.970","Text":"The r comes out front."},{"Start":"03:03.200 ","End":"03:08.005","Text":"What we end up with is the limit of"},{"Start":"03:08.005 ","End":"03:16.845","Text":"r times cosine cubed Theta plus sine cubed Theta."},{"Start":"03:16.845 ","End":"03:22.945","Text":"Now what we have here is a situation where as r goes to 0,"},{"Start":"03:22.945 ","End":"03:25.345","Text":"this bit goes to 0,"},{"Start":"03:25.345 ","End":"03:28.885","Text":"but this bit is bounded."},{"Start":"03:28.885 ","End":"03:33.250","Text":"Might remember that it\u0027s bounded because cosine of"},{"Start":"03:33.250 ","End":"03:39.535","Text":"Theta is between minus 1 and 1 or the absolute value is less than the 1."},{"Start":"03:39.535 ","End":"03:41.575","Text":"Similarly for sine Theta,"},{"Start":"03:41.575 ","End":"03:45.975","Text":"that\u0027s also at most equal to 1."},{"Start":"03:45.975 ","End":"03:50.005","Text":"Cosine cubed also can be between minus 1 and 1."},{"Start":"03:50.005 ","End":"03:55.910","Text":"The worst you could get here is up to 2 and down to minus 2, but it\u0027s bounded."},{"Start":"03:55.910 ","End":"04:01.735","Text":"The theorem is that something that goes to 0 times something bounded also"},{"Start":"04:01.735 ","End":"04:08.145","Text":"goes to 0 and so this limit is equal to 0."},{"Start":"04:08.145 ","End":"04:13.735","Text":"That\u0027s the left-hand side of this limit equals value, that\u0027s the limit."},{"Start":"04:13.735 ","End":"04:17.530","Text":"I\u0027ll just highlight it and now we\u0027ll go and do the other part,"},{"Start":"04:17.530 ","End":"04:22.730","Text":"the part that says value."},{"Start":"04:25.410 ","End":"04:30.180","Text":"Value means f of x,"},{"Start":"04:30.180 ","End":"04:31.620","Text":"y when x, y equals 0,"},{"Start":"04:31.620 ","End":"04:33.390","Text":"0 and we want f of 0,"},{"Start":"04:33.390 ","End":"04:35.340","Text":"0 and f of 0,"},{"Start":"04:35.340 ","End":"04:37.530","Text":"0 equals, here it is."},{"Start":"04:37.530 ","End":"04:40.410","Text":"When f is equal to 0,"},{"Start":"04:40.410 ","End":"04:44.920","Text":"when x, y is 0, 0 so this is equal to 0."},{"Start":"04:45.220 ","End":"04:48.215","Text":"Since 0 equals 0,"},{"Start":"04:48.215 ","End":"04:51.710","Text":"we have the limit equals value is confirmed."},{"Start":"04:51.710 ","End":"04:57.930","Text":"In other words, the function is continuous and we\u0027re done."}],"ID":8921}],"Thumbnail":null,"ID":4970}]

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