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By considering different paths of approach, show that the functions in Exercises $41-48$ have no limit as $(x, y) \rightarrow(0,0)$ .$$f(x, y)=-\frac{x}{\sqrt{x^{2}+y^{2}}}$$

$\lim_{y\to 0} f(x,y) = -1$, but, $\lim_{x\to y} f(x,y) = \frac{-1}{\sqrt{2}}$

Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

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with multi variable limits instead of approaching just from the left and the right of one value, as we did with single variables. We're going approach from many different directions of lines. So the one line that we could check is the line why you call zero or the approaching from the X axis? We could say so the limit as y approaches zero, would end up getting us, uh, negative X over square root of X squared would be X and then simplifying this, we'd end up getting negative one. Now, if we can get another value, we can prove that this limit does not exist. So, for instance, let's approach from the line. Why equals X or another way to say that is as X approaches. Why that would get us negative y on top, divided by the square root. Oh, uh, why squared plus y squared. Which one end up giving us two y squared? Taking the square root, we'd end up getting negative. Why over square root to why? And then simplifying these, we'd end up getting a fraction of negative one over route to And because we're getting two different values approaching from two different lines towards this, this 20.0 we know that this limit does not exist

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