Limit of function $f: \mathbb{R}^{2} \to \mathbb{R}$ and polar coordinates

Question

Let's say we have to check whether $$ \lim_{(x,y) \to (0,0)} \frac{x|y|}{x^{2} + y^{2}} = 0 $$Then if I pick $t > 0$ and consider the path $t \mapsto (t,t)$, then we get that the function above is $\frac{1}{2}$ on this path, and approaches $\frac{1}{2}$ as $t \to 0$. But suppose I consider stuff in polar coordinates, then isn't the above equation equivalent to showing $$ \lim_{(r, \theta) \to (0,0)} \frac{r\sin(\theta) |r\cos(\theta)|}{r^{2}} = 0 $$ and we see this indeed this $\to 0$ as $(r,\theta) \to 0$? Where am I going wrong?

Answer 1

$(x,y) \to (0,0)$ is not equivalent to $(r, \theta) \to (0,0)$. How do you get $\theta \to 0$ from $(x,y) \to (0,0)$?

You have to take limit as $ r \to 0$ and then you are left with $\sin \theta \cos |\theta|$. The fact that this limit depends on $\theta$ shows that the given limit does not exist.