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$$\lim_{(x,y)\to(0,0)} \frac{xy}{\sqrt{x^2+y^2}}$$

Approaching (0,0) along x or along y both result in the limit approaching 0, so you want to make sure that the limit exists by doing more tests.

My solutions manual uses $x = y^2$ (or $y = x^2$). Why either of those? Why not $y=x$ or $x=y$? Why a parabola?

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up vote 8 down vote accepted

None of those choices suffices to prove that the limit is $0$, so I don't know what the solution writer meant. No finite number of ways to approach $(0,0)$ can be enough to show that the limit exists. On the other hand, in a case where the limit doesn't exist at a point, one way to show it in some cases is to show that the function approaches different limits as you approach the point along two different curves.

In this case, $(|x|-|y|)^2\geq 0$ implies $|xy|\leq\frac{1}{2}(x^2+y^2)$, so that $$\left|\frac{xy}{\sqrt{x^2+y^2}}\right|\leq\frac{1}{2}\sqrt{x^2+y^2}.$$ This makes it easy to see that the limit is $0$.

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You're right. The author did use the squeeze theorem to prove that the limit existed. I was just wondering why he chose x=y^2 as one more test to see what the limit was. Why not x=y? or y=x? – ShrimpCrackers Mar 13 '11 at 8:18
@ShrimpCrackers: In that case, I guess the author was just trying to motivate the plausibility of its existence, and there's no definite rule for what to check in doing so. Checking $x=y$ as you suggest, you'd obtain a bit more informal evidence. As to why the author made that particular choice as a plausibility check, my answer is I don't know. My guess is that it was motivated by another problem where the limit doesn't exist, and considering $x=y^2$ or $y=x^2$ is used in showing this; force of habit, maybe. I don't see any mathematical reason. – Jonas Meyer Mar 13 '11 at 8:22
The other possibility is that the author wanted to use a less obvious choice. When I was taking and now when I teach calculus, after checking x=0 and y=0, most people tend to check y=x next, and maybe the author was just trying to encourage another option of curve to check. – Becca Winarski Mar 13 '11 at 13:53
@Becca Winarski: I like that answer. It's nice to have some variety. – Jonas Meyer Mar 13 '11 at 17:25

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