Find the limit of $\frac{\bar{z}}{z}$ as $z$ goes to $0$. I put it in exponential form to get $\dfrac{re^{-i \theta}}{re^{i \theta}}$ but I think I'll get $\frac{0}{0}$ which isn't defined and isn't a good enough proof to say it doesn't have a limit.

$$z=re^{it}\,\,,\,r>0\,,\,r\in\Bbb R\,\,,\,t\in [0,2\pi]\Longrightarrow\frac{\overline z}{z}=e^{-2it}$$
From the above it follows that the limit depends on the angle $\,t\,$ and, thus, it doesn't exist.
