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Define $f(0,0)$ in a way that extends$$f(x, y)=x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}}$$to be continuous at the origin.

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Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

Missouri State University

Harvey Mudd College

Baylor University

Boston College

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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02:56

Define $f(0,0)$ in a way t…

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04:40

Continuous extension Defin…

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For the function$$f(x,…

So first question is asking us to find F. Zahra's air in a way that extends have to be continuous at the origin. So in order for a function F B contains of the origin, the function value at the origin must be equal to the limit of the function as X y approaches 00 So, in other words, we wanted to find at 00 in order for it to be contained with origin, it has to be equal to the limit. Is X y those 200 of my function? So to go about this problem, I'm going to go ahead and find this limit of this function. So I think lim as X y goes to 00 three X squared. Why over X squared plus y squared here. I do notice if I plug in 00 into this directly, I'm going to get it and determine it form. So I'm gonna go ahead and use the trick that we use in the previous block of questions as to substituting my ex and why in for the polar coordinates? So in sub student, the Axis eagles are co sign of data wise Eagle, that are sinus data. And so then I get the limit. It's now our approaches. Zero ofthree r squared co sign squared of data Times are sinus Ada all over our squared co sign squared US data plus R squared sine squared data. And so simplifying that out on the top i c I can multiply these ours together. So I get three are cubes co sign squared of state A sinus data No, on bottom, I notice I can factor on our square and I'm left with co sign squared Fate A plus sign squared data which, as he saw quite often and the previous problems that's just equal to one son just left with an r squared on bottom, there s a limit. And so then I noticed this r squared cancels of our cute nun slept with one are here. So I have the limit as our approaches zero of three are go sign square to state a time sign of data which now, when I plug in our equal zero into here, I get that the limit is equal to zero. So therefore we're into fine F 00 equals zero that extends actually continuous at the border

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