TITLE: L'hospital rule for two variable.
QUESTION [25 upvotes]: How to use L'hospital rule to compute the limit of the given function 

$$\lim_{(x,y)\to (0,0)} \frac{x^{2}+y^{2}}{x+y}?$$

REPLY [14 votes]: The answer by Christian Blatter is correct.  This particular limit is undefined.
I have recently written a paper giving a l'Hospital's rule for many types of multivariable limits.  You can find it at
http://arxiv.org/abs/1209.0363