<s>[INST] Calculate $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz+zx}{x^2+y^2+z^4}$ $$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz+zx}{x^2+y^2+z^4}$$
In order to calculate this limit, I did:
$$\lim_{(x,y,z)\to(0,0,0)}\left(y\frac{x}{x^2+y^2+z^4}+z\frac{y}{x^2+y^2+z^4}+x\frac{z}{x^2+y^2+z^4}\right)$$
then, in each of these terms, one part is limited by $1$, and the other goes to $0$, then the entire thing goes to $0$
Am I rigth?
 [/INST] Let $y=nx$, $z=mx$. So,
$$
\frac{xy+yz+xz}{x^2 + y^2 + z^4} = \frac{n+nm+m}{1 + n^2 + m^4x^2}.
$$
Limit depends on $n$ and $m$; so, it doesn't exists.
</s>