TITLE: Is there ever a case where a two-dimensional limit exists but not the one dimensional limits?
QUESTION [5 upvotes]: I'm just learning some proof-based multivariable calculus, and in Apostol Volume 2 chapter 8.5 exercise number 2 there is the following statement.
If 
$$
\lim_{(x,y)\to (a,b)}f(x,y)=L
$$
and $\lim_{x\to a}f(x,y)$ and $\lim_{y\to b}f(x,y)$ both exist, prove that 
$$
\lim_{x \to a}[\lim_{y \to b}f(x,y)] = \lim_{y \to b}[\lim_{x \to a}f(x,y)]=L.
$$
This is easy enough to believe. My question is whether it is redundant to state that the single variable limits exist, given that the two variable limit exists. Is there ever a case where the two dimensional limit exists but the individual one dimensional limits don't? Thanks.

REPLY [4 votes]: Consider 
$$f(x,y)=\begin{cases} -x, & y\in\Bbb Q \\ x, & y\notin\Bbb Q \end{cases}.$$
Then  $\lim\limits_{y\to 0} f(x,y)$ does not exist when $x\ne 0$, but $\lim\limits_{(x,y)\to (0,0)} f(x,y) = 0$.