How to compute the $\lim_{(x,y)\to(0,0)}\frac{x^{2}}{x^{2}+y^{2}}$? I have computed the limite of $\lim_{(x,y)\to(0,0)}\frac{x^{2}}{x^{2}+y^{2}}$ as follows: 



I want to know if with the first two paths -Path 1 and Path 2-, are sufficient to say that the limit does not exist, because the given paths are different -$1 \neq \frac{1}{2}$.

What about the third one?

Is this procedure correct?
Yes, it is enough to conclude from the first two paths that the limit does not exist.
Yes. This means as you get arbitrsrily close to the origin, you have some function values that do not get arbitrarily close to any potential limit (because they get arbitrarily close to some other potential limit)