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