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Find the limit of $f$ as $(x, y) \rightarrow(0,0)$ or show that the limit does not exist.$$f(x, y)=\cos \left(\frac{x^{3}-y^{3}}{x^{2}+y^{2}}\right)$$

1

Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

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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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02:41

Find the limit of $f$ as $…

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Find the limit, if it exis…

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even this function that affects wise equal to co sign of X Q minus y cute over X scripless y squared and refer pacifying limit of F as X y approaches 00 or show the limit is nine of this. So before you go into the sub problems, it goes into some detail about how you can change variables of polar coordinates, and sometimes that's easier to find a limit. So that kind closing in. That's what you want to do here. So point Thanks. Remember, is that X for polar cornets is equal to our coastline of data and that why is equal to our sign of data. So I'm going to go ahead and switch this the polar coordinates. And when I do that in taking a limit again, I'm taking a limit as our approaches zero as he saw in that section of co sign of Gan accessing our co science data. So if I'm keeping that get r cubed Poseidon que dis ada minus why cube such man are acute Sign Cuba Zeta all over r squared co signs great of data plus r squared sine squared sit and said, I'm trying to do some simplifying here. So my first almost on the bottom I can factor on our squared. And why did I'm left with co sign squared plus science weird which, as we know all know that's just equal to one. So on the bottom, just left with are squared on the top of the news Something online see factor on r cubed and I'm just left with co sign que dis data minus sign Cubist Dada And so a notice I came Cancel out more These are stressing and be gone Some slept with an r on tops like it Limit Starbuck zero of co sign of our times Co sign Cuba's data minus sign Cuba's data. And so now I want to put in our equals zero and two Here I just get coastline of zero, which was equal to one

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