Double limit

What we can do is make a substitution. We let $x=r\cos(\theta)$ and $y=r\sin(\theta)$ .

Now instead of writing $\lim_{x,y \to 0,0}$ we can replace it with $\lim_{r\to 0}$ .

$\large \lim_{x,y \to 0,0} \frac{xy}{x^2+y^2}= \lim_{r \to 0}\frac{r^{2}\cos(\theta)\sin(\theta)}{r^2}= \lim_{r \to 0}\cos(\theta)\sin(\theta)=\cos(\theta)\sin(\theta)$

Since the limit isn't constant (i.e depends on $\theta$ ) we say the limit does not exist.

Along the line $x=0$ the value is 0, but along the line $x=y$ the value is $\frac{1}{2}$ , so that this limit fails to exist.