Find $\lim\limits_{(x,y) \to(0,0)} \frac{xy^3-yx^3}{ x^4 + y^4}$ Find a limit or prove that it doesn't exists: 
$\lim\limits_{(x,y) \to(0,0)} \frac{xy^3-yx^3}{ x^4 + y^4}$
I tried to prove that it doesn't exist by finding two sequences with different limits, but I couldn't find them, so I suppose the limit exists, but unfortunately I don't have any idea how to prove it.

Consider the limit as $y=ax$, where $a$ is a constant. Then 
$$\lim_{x\to 0}\frac{x^4a^3-x^4a}{x^4+a^4x^4}=\frac{a^3-a}{1+a^4}.        $$
