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By considering different paths of approach, show that the functions in Exercises $35-42$ have no limit as $(x, y) \rightarrow(0,0) .$$$f(x, y)=-\frac{x}{\sqrt{x^{2}+y^{2}}}$$Graph cannot copy

the function has no limit at $(x, y) \rightarrow(0,0)$

Calculus 3

Calculus 1 / AB

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

Applications of the Derivative

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in this problem, consider the following function. F x y is Cipel's toe minus X over under Root X Square plus Vice Square. Now the object is to show that the function has no limit, as ex wife approaches 00 by considering different parts off approach. Now, if the left and right limits exist at the point and are equal then on Lee, the limit off the function at that point exist. Otherwise, it does not exist. Now, along the part that is by is equals, Toe X and X is greater than zero. The limit off the function that is X Y approaches 00 will be limit. X Y approaches 00 minus eggs over under root X square, plus by square, the equal toe limit X approaches. Zero positive. And why is it was toe X minus X or X square plus X square. It should be equal toe limit. X approaches zero positive minus X Our under two X square equals toe limit. X approaches zero positive minus X over. I know two more eggs now Use more X is equals to eggs. You X is greater than zero than we will get. Limit X approaches. Zero positive minus X or under two X, which will be equal toe limit X approaches zero positive minus one over under two and minus one or under. Rule two. Now along the path that will be by physicals. Toe eggs on X is less than zero. We will get limit X Y approaches 00 minus eggs or under rude X squared plus y square is equals toe limit X approaches. Zero negative and why is equals to eggs minus X over. Under Ruth, X square plus X square is equals toe limit X approaches zero negative minus X over under two X square is equals toe limit X approaches zero negative minus eggs. Our under two more eggs now use more exes equals toe eggs. If X is less than zero, then we will get Lim X approaches the O negative minus eggs under two minus eggs. We will get limit. X approaches the O negative one hour Who to which will be one hour route to now use the two results. It confirms that both results are defend, so the limit off the function does not exist at X y approaches 00 Therefore, the function has no limit at X y approaches 00 So that's a solution

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By considering different paths of approach, show that the functions in Exerc…

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