Does the following equality generally hold?
$ \lim_{x\to\infty, y\to\infty} f(x, y) = \lim_{z\to\infty} f(z, z) $
If not, what are the necessary conditions for the above equation to hold?
Does the following equality generally hold?
$ \lim_{x\to\infty, y\to\infty} f(x, y) = \lim_{z\to\infty} f(z, z) $
If not, what are the necessary conditions for the above equation to hold?
Consider $f(x,y)=e^{-(x-y)^2}$. Then each of the iterated limits is $0$ at infty, and obviously $f(z,z)=1$ for all $z$.
To make the equality hold, you need to change $\lim_{x \to \infty}\lim_{y \to \infty} $ to $\lim_{x,y \to \infty}$. Then, if $f$ has a limit at $\infty$ the equality $\lim_{(x,y) \to \infty}f(x,y)=\lim_{z \to \infty}f(z,z)$ holds.