He guys, I need to show if the following limit exists,
$$
f(x,y)=\frac{x^3-xy^3}{x^2+y^2},
$$
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?
Let $f: X \to X$ be a map of sets and $\text{Sym}_X$ be the group of permutations of $X$. Then if $f$ commutes with every permutation $\sigma \in \text{Sym}_X$ what else can we say about $f$?
We can say that if $|X| \gt 1$, then $f$ is not constant since if:
$X= \{a,b,c\dots\}$ and $f(X) = c$....
