$\textbf{Question}$: Find the limit of each function at the given point, or explain why it does not exist: $f(z) = (z-2)log|z-2|$ at $z_0 = 2$.
$\textbf{Answer}$: Since $z_0= 2$, let $z' = z-2$.
$$\lim_{z\to z_0} f(z) = \lim_{z' \to 0} f(z) =z'\log |z'|$$
$$=z' (\ln |z'| +i\cdot 0) = \frac{ln |z'|}{\frac{1}{z'}}$$
By L'Hôpital's Rule, $$= \frac{\frac{1}{z'}}{-\frac{1}{(z')^2}}=\lim_{z'\to 0} (-z') = 0$$.
Therefore, the limit of the function is 0 at $z_0=2$.
