# Finding limit of function with more than one variable

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

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?

-

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 $0$.
@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