I'm trying to calculate the following limit:
$$\lim_{(x,y)\to (0,0)} \frac{x(1+2x/y)}{(1+x/y+(x/y)^{2})^{2}}.$$
When I use WolframAlpha, it tells me the limit doesn't exist. However, if I change to polar coordinates I get
$$ \begin{equation*} \lim_{(x,y)\to(0,0)}\frac{x(1+2x/y)}{(1+x/y+(x/y)^{2})^{2}} = \lim_{r\to 0^{+}}\frac{r\cos(\theta)(1+2\cot(\theta))}{(1+\cot(\theta) + \cot^{2}(\theta))^{2}} = 0 \end{equation*} $$
Can someone tell me what's going on here?