Non-existence of $\lim \limits_{(x, y) \to (0,0)} \frac{x^3 + y^3}{x - y} $ 
How to show that $\lim \limits_{(x, y) \to (0,0)} f(x, y)$ does not
  exist where,   
$$f(x, y) =  \begin{cases} \dfrac{x^3 + y^3}{x - y} \; ; & x \neq y \\
 0 \; \;\;\;\;\;\;\;\;\;\;\; ; & x = y  \end{cases}  $$

I tried bounding the value of the function as $(x, y)$ approaches $(0,0)$ but was not successful. I graphed the function and saw that it was actually approaching $0$ but Microsoft Math failed to render some points around the $x = y$ plane. So I imagine I need to find points $(x, y)$ such that $x$ is extremely close to $y$ but $(x, y)$ is not as close to $(0,0)$ to show that the function does not approach $0$ around the origin. But was unable to think of any. 
Hints or suggestions would be awesome. Any help is appreciated.

Hint:  What happens on the curves $$(x(t), y(t)) = (1/t \pm e^{-t}, 1/t), \quad t > 0?$$
Here is a little animation of the surface in question.  It was surprisingly challenging to obtain a smooth plot in the neighborhood of $(0,0)$, but I managed to find a nice parametrization.  The red and green curves are the ones I described, but it is quite clear that other paths to the origin will also work.

