I want to calculate $$\underset{z\rightarrow \infty }{\lim} \, \frac{z}{\log z}=\underset{z\rightarrow \infty }{\lim} \,\frac{z}{\log|z|+ i\operatorname{Arg} z} $$
but I have some problems
First $\underset{z\rightarrow \infty }{\lim} \,\operatorname{Arg}z$ doesn't exist, that means that the denominator is $\infty$ because of the $\log |z|$?
If we don't have a problem with the denominator and its just $\infty $ how can I solve the $\frac{\infty}{\infty}$ problem
I tried to take the norm $| \,\frac{z}{\log|z|+ i\operatorname{Arg} z} |<\frac{|z|}{|\log|z|-i\pi|}$ or to show that the limit doesn't exist by taking 2 subsequences ect, but it didn't help me solve the $\frac{\infty}{\infty}$ problem.
Any hints?