Iterated Limits

Question

If $$\lim_{(x,y) \to (a,b)} f(x,y) = L$$ and if the one dimensional limits - $$ \lim_{x \to a} f(x,y)$$ and $$ \lim_{y \to b} f(x,y)$$ both exist, then prove that -
$$ \lim_{x \to a}\left[\lim_{y \to b} f(x,y)\right] = \lim_{y \to b}\left[\lim_{x \to a} f(x,y)\right] = L$$

Answer 1

Counterexample: $f(x,y)=r^2\theta$ where $r$ and $\theta$ are the polar coordinates. The latter runs in $[0,2\pi)$. Then $f$ converges to zero continuously in the origin, but only one of the iterated limits exists in a neighborhood of zero.