Is $\lim_{z\rightarrow 1}\frac{z_0}{\log(\frac{1}{z-1})}=0$ where $z_0$ is a constant.
I think it is but I don't know how to prove it rigorously.
Also when the questions mentions the complex logarithm is it referring to the multivalued function or the principal logarithm?
My Reasoning:
I know that $|\log(\frac{1}{z-1})|=|\ln(|\frac{1}{z-1}|)+i\arg(\frac{1}{z-1})|$. As $z\rightarrow 1$, $\ln(|\frac{1}{z-1}|)\rightarrow \infty$
Hence, $\lim_{z\rightarrow 1}|\frac{z_0}{\log(\frac{1}{z-1})}|=0$
Is this true?