Show that $\lim\limits_{(x,y)\to(0,0)}\frac{x^3y-xy^3}{x^4+2y^4}$ does not exist. 
Show that $$\lim_{(x,y)\to(0,0)}\frac{x^3y-xy^3}{x^4+2y^4}$$ does not exist.

I'm not even sure how to approach this. I tried factoring out $xy$ in the numerator to get $xy(x^2 - y^2)$, but I don't think that gets me anywhere with the denominator. 

Doing the change to polar coordinates:
$$
\frac{x^3y - xy^3}{x^4 + 2y^4} =
\frac
{r^3\cos^3\theta\,r\sin\theta - r\cos\theta\,r^3\sin^3\theta}
{r^4\cos^4\theta + 2r^4\sin^4\theta}
= \frac
{\cos^3\theta\sin\theta - \cos\theta\sin^3\theta}
{\cos^4\theta + 2\sin^4\theta},
$$
dependent of $\theta$ (and independent of $r$), so the limit does not exists.
