lim([(0,y) -> (0,0)] 0/(0 + y^2) = lim(y -> 0) 0 = 0
Then, consider the lim along the path y = 0; thus, we have
lim([(x,0) -> (0,0)] 0/(x^2 + 0) = lim(x -> 0) 0 = 0
Although the limits along the above two paths are the same does not necessarily mean that a limit infact exists. In order for a lim to exist, the lim must be the same along every path through (0,0), not just the two picked. Looking at the path y = x, we have:
lim([(x,x) -> (0,0)] (x*x)/(x^2 + x^2) = lim(x -> 0) x^2/(2x^2) = 1/2
And thus, since the limit along the above path does not match the first two paths, the limit does not exist.