I'm having some problems resolving this limit. The fact is that I'm finding little information about limits in three variables online and in my manuals, and even less information about three variables approaching infinite. This is (one) of the limits:
$\lim_{x,y,z\to \infty}\frac{1+xy}{xz^2}$
My idea is to split the fraction in two:
$=\lim_{x,y,z\to \infty}\frac{1}{xz^2}+\lim_{x,y,z\to \infty}\frac{y}{z^2}$
Now the first one obviously goes to zero but I'm not sure about the second part going to zero too.
We have that
for $z^2=y \implies \frac{1+xy}{xz^2}=\frac{1+xy}{xy} \to 1$
for $-z^2=y \implies \frac{1+xy}{xz^2}=\frac{1+xy}{-xy} \to -1$