MAT334--2020F > Quiz 2

Quiz 2 f(z)=(z-2)\log|z-2| at z_0=2

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kaye:
Problem(3pt). Find the limit of each function at the given point, or explain why it does not exist.
$$f(z)=(z-2)\log|z-2| \text{ at } z_0=2$$

Let $z'=z-2$, then
\begin{align*}
\lim_{z \to \infty} |f(z)| &=\lim_{z' \to 0} |f(z)|\\
&= \lim_{z' \to 0} |z'\log|z'||\\
&= \lim_{z' \to 0} \frac{\log|z'|}{\frac{1}{|z'|}}
\end{align*}
When $z' \to 0$, we get $\frac{\infty}{\infty}$, now use the L'Hospital's Rule we have:
\begin{align*}
\lim_{z \to \infty} |f(z)| &= \lim_{z' \to 0} \frac{\log|z'|}{\frac{1}{|z'|}}\\
&= \lim_{z' \to 0} \frac{\frac{1}{|z'|}}{\frac{-1}{|z'|^2}}\\
&= \lim_{z' \to 0} - |z'|\\
&= 0
\end{align*}
The limit of $f(z)$ at $z_0=2$ is 0.