limit of modulus of z over z approach infinity in complex plane

Question

I am solving $$\lim_{z \rightarrow \infty} \frac{|z|}{z}$$, for now I have \begin{align*} \lim_{z \rightarrow \infty} \frac{|z|}{z} &= \lim_{z \rightarrow \infty} \frac{\sqrt{z\bar{z}}}{z} \\ &=\lim_{z \rightarrow \infty} \sqrt{\frac{\bar{z}}{z}} \\ &=\sqrt{\frac{e^{-i\theta}}{e^{i\theta}}} \\ &=e^{-i\theta} \end{align*} Does this looks right? Thanks.

Answer 1

Looks right, as a start. Looks actually like the limit doesn't exist, since by taking $z=re^{i\theta}$ and letting $r\to \infty$ we get different limits for different $\theta$.