<s>[INST] Show $\lim_{(x,y)\to(0,0)}\frac{x^3-xy^3}{x^2+y^2}=0$ I need to show if the following limit is exists (and exists $0$),
$$
f(x,y)=\frac{x^3-xy^3}{x^2+y^2},
$$
for $\vec x\to 0$. I tried out the following:
$$
\frac{x^3-xy^3}{x^2+y^2}\leq\frac{x^3-xy^3}{x^2}=1-\frac{y^3}{x^2},
$$
but obviously I'm stuk here. A similar approach by omitting $y^2$ gives the same problem. What could I do next?
 [/INST] Note that
$$\left|x^3-xy^3\over x^2+y^2\right|\le|x|\left|x^2\over x^2+y^2\right|+|xy|\left|y^2\over x^2+y^2\right|\le|x|+|xy|$$
</s>