Calculus 3 Lecture 13.2: Limits and Continuity of Multivariable Functions (with Squeeze Th.)

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Intro to multivariable limits: 0:00 Showing a limit (with two variables) does not exist: 26:54 Showing a limit (with three variables) does not exist: 1:12:19 Showing a limit does exist: 1:32:07 changing to polar coordinates: 1:32:28 Continuity (domain and compositions): 1:41:47 Summary of computing limits: 2:06:56 - 2:07:24 Squeeze theorem (converting to inequalities): 2:07:26

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conceptual talk until 00:00 Examples of 2 dependent variables: 24:00 Examples of 3 dependent variables: 1:12:15 Squeeze theorem : 2:07:25

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Independent*

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Thank you for showing how to apply Squeeze Theorem to these problems!!! My teacher tried to explain it to us in half of a lecture and you did it under 10 minutes in a way that I can understand!

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I have been struggling with limit continuities for weeks, this video's first ten minutes cleared up everything I had wrong!

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Your classes are helping me get through this multivariable stuff during these trying times. Thank you!

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My professor for Calc 3 is named Hongwei and speaks with a hard accept from china so when he goes over anything its always a bigger challenge to not only understand the concept but him as well. So when he went over this section he sped through it only talking about the epsilon-delta definition of it and then using polar to solve some problems, never fully explaining the actual concept. This video has not only saved my grade but also my understanding of limits for all future calculations I'll be doing in my career as an engineer. Thank you so much Professor Leonard!

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[@georgesadler7830]

Professor Leonard, thank you for another well organized video/lecture on Limits and Continuity on Multivariable Functions with the classic Squeeze Theorem in Calculus Three. These concepts are introduced in Calculus One, which improve my understanding of the subject in Multivariable Calculus.

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+Brad Nissen I'm releasing them as fast as I can make them, I hope they will help. Good luck!

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+sandile mabika He actually already has released it, but the video was shot with a 4-D device in 4-D setting so unfortunately 3-D people are unable to see it. Maybe you should get 4-D glasses, I am selling if you are interested.

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Could you also give more examples of proving the limit DOES exist in different ways and what methods to use when?

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Thank you so much! I'm surprised my book never mentions the squeeze theorem and instead goes through a bunch of other non-sensical stuff... You made it all perfectly clear, and the theorem kind of makes sense intuitively as well.

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[@gmcenroe]

In this first and second example if you do a 3D plot (Used Mathematica in my case) you can see that these functions are not continuous at (0,0) but has a cusp along one axis at the point (0,0) in example 1 and a cusp along the -x,y and x.y diagonals in example 2 therefore the limit is undefined. In the example lim as (x,y)->(1,0) of (2xy-2y)/(x^2 +y^2 -2x +1)Professor Leonard found that the lim as y->0 of 2y^2/2y^2 =0 presumably because this reduced to the lim y->0 of 2y^2/2y^2 is 0/0 =1 and concluded that the limit was undefined. However Mathematica finds this limit of the base limite to be 0, so the original problem has a limit of 0 also which is also what Mathematica found. This is in agreement with the plot of the surface and examining the point (x,y) = (1,0) Mathematica's determination of the original limit to be 0, so I believe his answer is differant. I guess this depends on your definition of 0/0 or it could be a bug in Mathematica. Please comment Professor Leonard and others if you can. Time point 1:11:48.

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[@tomasvenegas9600]

i think you might be wrong here. When you use for example polar coordinates and you get a number it's not strong enough to prove that the limit exists, you'd have to use that number as a candidate of a posible value of the limit and then plug it the squeeze theorem or the epsilon-delta definition. Then just then you can prove the limit exists.

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[@keldonchase4492]

0:00 - Introduction to limits and continuity of multivariable functions 0:59 - Review of limits from single variable calculus 6:23 - Limits of multivariable functions 19:05 - How to prove a limit doesn't exist 24:08 - Example 1 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis and the y-axis) 44:14 - Example 2 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis, the y-axis, and a linear function) 54:17 - Example 3 (limit of function with two independent variables) (showing the limit doesn't exist going along the x-axis, the y-axis, and a cubic-root function) 1:02:30 - Example 4 (limit of function with two independent variables) (showing the limit doesn't exist going along different paths) 1:12:19 - Limits of functions with three independent variables 1:14:39 - Example 5 (limit of function with three independent variables) (showing the limit does't exist going along different paths) 1:20:51 - Example 6 (limit of function with three independent variables) (reparameterizing the limit: assigning "t" expressions to the independent variables) 1:24:18 - Example 7 (limit of function with two independent variables) (plugging in values to get direct answer) 1:27:22 - Example 8 (limit of function with two independent variables) (plugging in values to get direct answer) 1:29:19 - Example 9 (limit of function with three independent variables) (plugging in values to get direct answer) 1:31:54 - Example 10 (limit of function with two independent variables) (utilizing polar identities to evaluate limit) 1:37:22 - Example 11 (limit of function with two independent variables) (using polar identities and L'Hopital's Rule to evaluate limit) 1:41:48 - Introduction to Continuity in Multivariable Calculus 1:43:53 - Definition of Continuity in Multivariable Calculus 1:49:22 - Example 12 (continuity of function with two independent variables) 1:50:26 - Example 13 (continuity of function with two independent variables) (determining domain of function) 1:51:35 - Example 14 (continuity of function with two independent variables) (determining domain of function) 1:53:33 - Example 15 (continuity of function with three independent variables) (determining domain of function) (domain includes all points except the ones that form that particular SPHERE) 1:55:47 - Introduction to composition problems doing example 16 1:58:40 - Example 17 (composition problem setting "t" equal to f(x,y)) (domain restrictions with g(t)) 2:01:55 - Example 18 (composition problem) (domain restrictions with f(x,y)) 2:04:59 - Introduction to Squeeze Theorem 2:06:30 - Example 19 (Squeeze Theorem)

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