Multivariable Limit of Fraction: $\lim_{x,y \to (0,0)}\frac{x^4+y^4}{x^4 +(x+\sqrt[3]{y})^2+y^4}$ I am supposed to find the following limit, which seems to be zero, but I am having trouble proving it.
$$\lim_{x,y \to (0,0)}\frac{x^4+y^4}{x^4 +(x+\sqrt[3]{y})^2+y^4}$$

One has, as $(x,y) \to (0,0)$, with $x=-\sqrt[3]{y}$,
$$
\frac{x^4+y^4}{x^4 +(x+\sqrt[3]{y})^2+y^4}=\frac{x^4+y^4}{x^4 +0+y^4} \to1
$$ and one has, as $(x,y) \to (0,0)$, with $x=0$,
$$
\frac{x^4+y^4}{x^4 +(x+\sqrt[3]{y})^2+y^4}=\frac{y^4}{y^{2/3}+y^4}=\frac{1}{\frac1{y^{10/3}}+1} \to 0
$$  the limit can't exist.
