The limit may very well not exist.  Always test for this
possibility first.

Let's test for this:

If the value of the limit is not the same for all possible 
approaches, or paths, to (0,0), then the limit does not exist.

Let's choose the path y=kx for some constant k.
So we substitute kx for y:



 







When x approaches 0, this approaches

  

which has different values for different choices of k.
For instance, it's 0 if k=0 and 1/2 if k=1, and 2/5 if k=2, etc.

That means the limit cannot exist, for it must approach
the same value regardless of the path taken to the origin.
But as we see this is not the case.

Answer:  the limit does not exist.

Edwin