Question: Find the limit of each function at the given point, or explain why it does not exsit.
f(z)=(1−Im z)^{-1} at z_{0}=8
and then at
\begin{equation*}
z_{0}=8+i.
\end{equation*}
Answer:
- When z_{0} = 8, we have $$\lim_{z\to 8}f(z)=\lim_{z\to 8}\frac{1}{1- Im[8]} = \lim_{z\to 8}\frac{1}{1-0} = 1$$
- When z_{0} = 8+i, we have $$\lim_{z\to 8+i}f(z)=\lim_{z\to 8+i}\frac{1}{1- Im(8+i)} = \lim_{z\to 8+i}\frac{1}{1-1}$$ Since the denominator cannot be zero, hence when z_{0} = 8+i, the limit does not exist.