Proving $\lim_{(x,y)\to (0,0)} \frac{x^{3}-y^{3}}{x^{2}+y^2}=0$

Question

Let's say I wish to find the limit of $\frac{x^{3}-y^{3}}{x^{2}+y^{2}}$ as $(x,y)\to (0,0)$. It is very simple to just let $y=mx;m\in\mathbb{R}$, and speculate that since the expression comes out to be $0$ on taking the limit and not a function of $m$, the limit may be $0$. Although, this obviously isn't very rigorous, because we're missing on infinitely many other paths of approach. So how would we prove this limit using Epsilon-delta. Any hints or ideas are appreciated. Thanks.