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 $0$.