How to calculate $\lim_{(x,y) \rightarrow (1,-1)} \frac{x^2 - y^2}{x^3 + y^3}$ $$\lim_{(x,y) \rightarrow (1,-1)} \frac{x^2 - y^2}{x^3 + y^3}$$
By approaching along x = 1 or y = -1 and using L'Hopitals rule, I can establish that if the limit does exist, then it is $\frac{2}{3}$.
This matches the result given by Wolfram Alpha and is consistent with a 3D graph of the function.
However I am struggling to prove that this limit exists along all approach paths.
I have tried $x = -y$ and the Sandwich Theorem but still seem no closer to a general proof.
Can you give me a hint for to the correct method for proving that this limit exists?

\begin{align}
\lim_{(x,y)\to (1,-1)}\frac{x^2-y^2}{x^3+y^3}&=\lim_{(x,y)\to(1,-1)}\frac{(x+y)(x-y)}{(x+y)(x^2-xy+y^2)}\\
&=\lim_{(x,y)\to(1,-1)}\frac{x-y}{x^2-xy+y^2}\\
&=\frac{1-(-1)}{1-1\cdot(-1)+1}\\
&=\frac{2}{3}
\end{align}
