What is the limit of $\lim_{(x,y)\to (0,0)} \frac{x^2+y^2}{y}?$

Question

I am somewhat rusty with the limits when changing polar coordinates and the following question has arisen.

What is the limit of $$\lim_{(x,y)\to (0,0)} \frac{x^2+y^2}{y}?$$

In Wolfram, $\lim_{(x,y)\to (0,0)} \frac{x^2+y^2}{y}=0$ but, in polar coordinates, with $\theta=r$ $$\lim_{r\to 0, \theta=r} \frac{r}{\sin(\theta)}=\lim_{r\to 0} \frac{r}{\sin(r)}=1$$.

The limit not exists? Thanks.

Answer 1

Approach the limit along two different paths; $y=x$ and $y=x^2.$ Can you complete now? Even in polar version also the limit value varies with the different values of $\theta!$