Changing the order of limits

Question

I'm looking for an example of a function $f(x,y)$ such that $$\lim_{x\to a}\{\lim_{y\to b} f(x,y)\}\neq \lim_{y\to b}\{\lim_{x\to a}f(x,y)\}$$

Answer 1

Because it fails to hold for some $f(x,y)$!!

Consider $f(x,y)=\dfrac{x+y^2}{x+y}.$

$\lim_{x\to 0}\left(\lim_{y\to 0}f(x,y)\right)=1$ and $\lim_{y\to 0}\left(\lim_{x\to 0}f(x,y)\right)=0$