TITLE: Limit of complex function
QUESTION [8 upvotes]: Im trying to find the limit of:  
$$ \frac{\operatorname{Re}(z) \operatorname{Im}(z)}{z^2}$$
as z tends to zero.

REPLY [5 votes]: Another approach: use polar coordinates
$$z=re^{it}\,\,,\,0\leq t\leq 2\pi\Longrightarrow Re(z)=r\cos t\,\,,\,Im(z)=r\sin t\,\,,\,\text{and}\,\,\,z\to 0\Longleftrightarrow r\to 0 \Longrightarrow$$
$$\Longrightarrow \frac{Re(z)Im(z)}{z^2}=\frac{r^2\cos t\sin t}{r^2(\cos^2 t-\sin^2 t+2i\cos t\sin t)}=\frac{1}{2}\frac{\sin 2t}{\cos 2t+i\sin 2t}\xrightarrow [r\to 0]{} \text{doesn't exist}$$
as it depends on the angle $\,t\,$