L'hospital rule for two variable. How to use L'hospital rule to compute the limit of the given function 


  $$\lim_{(x,y)\to (0,0)} \frac{x^{2}+y^{2}}{x+y}?$$
There is no L'Hopital's Rule for multiple variable limits. For calculating limits in multiple variables, you need to consider every possible path of approach of limits.  

What you can do here:

Put $x=r\cos\theta$ and $y=r\sin\theta$, (polar coordinate system) and $(x,y)\to (0,0)$ gives you the limits $r\to 0$ and no limits on $\theta$.

Now we need to substitute these in your problem which then becomes $$\lim_{r\to 0}\dfrac{r^2}{r(\cos\theta+\sin\theta)}=\lim_{r\to 0}\dfrac{r}{(\cos\theta+\sin\theta)}$$ Now for paths  where $\cos\theta\to-\sin\theta$ or $\cos\theta=-\sin\theta$, the denominator $\to0$ or $=0$ while the numerator is just tending to $0$ but not exactly $0$.Since the second case exists $\implies$ the limit doesn't exist.

If extra condition that $(x+y)\neq 0$ is applied, even then the limit does not exist as for path $\theta=0$, limit is $0$ while for path $\sin\theta=-\cos\theta+r\cos^2\theta$, limit is something which can be non-zero.
Since there are two variables, the equation is a surface (2 dimensional) therefore, the limit is also a 2 dimensional patch. You should apply hopital rule in both directions, but since the equation is symmetric, that shouldn't be a problem.