If $\lim_{x\to\infty} \lim_{y\to\infty} f(x,y)$ exists, does $\lim_{y\to\infty} f(x,y)$ for $x$ sufficiently large?

Question

Let $f:\mathbb{R}^2\to [0,1]$ and assume that $\lim_{x\to\infty} \lim_{y\to\infty} f(x,y)$ exists. My question is: can I conclude that for $x$ sufficiently large, $ \lim_{y\to\infty} f(x,y)$ exists as well?

Answer 1

If you assume that $\lim_{x\rightarrow \infty} \lim_{y\rightarrow \infty} f(x,y)$ exists, then $\lim_{y\rightarrow \infty} f(x,y)$ exists, otherwise the first limit is not defined. I think you want $x,y$ to go to infinity at the same time. If this is the case, then it is wrong. A counterexample would be $$ f(x,y)= e^{-x^2} \cos(y)^2$$

Answer 2

Note that, since you are writing about $\lim_{x\to\infty} \lim_{y\to\infty} f(x,y)$, you are actually assuming that $\lim_{y\to\infty} f(x,y)$ exists if $x$ is large enough.