On epsilon-delta proofs for multi-variable limits

Question

Question

Compute the following limit $$\lim\limits_{(x,y)\to(0,0)} \frac{y^2}{x-y}.$$

My Answer

I was taught to first show that the limit along any line and curve through the origin is $0$ which is true in this case, so I begin to suspect that the limit does exist and is equal to $0$.

Now, let $\epsilon > 0$. We want to find $\delta > 0$ such that, if $$0 < \sqrt{x^2+y^2} < \delta,$$ then $$\left\lvert \frac{y^2}{x-y} - 0 \right\lvert < \epsilon.$$ In other words, if $$0 < \sqrt{x^2+y^2} < \delta,$$ then $$\frac{y^2}{\lvert x-y \lvert} < \epsilon.$$

However, I am stuck here. How should I proceed with my epsilon-delta proof?

Any intuitive explanations will be greatly appreciated!

Answer 1

This limit does not exist. If you take $y=\frac 1 n$ and $x=\frac 1 n+\frac 1 {n^{2}}$ and let $n \to \infty$ you get $1$.

Answer 2

Your "the limit along any line and curve through the origin is 0 which is true in this case" was wrong: take the curve $x=y+y^3.$