Question: Find the limit of the function at the given point, or explain why it doesn't exsit.
f(z)=(1−Imz)^{-1} at z_{0}=8 and then at z_{0}=8+i.
Answer:
- When z_{0} = 8, $$\lim_{z\to 8}f(z)=\lim_{z\to 8}(1- Im[8])^{-1} = \lim_{z\to 8}\frac{1}{1-0} = 1$$
- When z_{0} = 8+i, $$\lim_{z\to 8+i}f(z)=\lim_{z\to 8+i}(1- Im[8+i])^{-1} = \lim_{z\to 8+i}\frac{1}{1-1}$$, since the denominator cannot be zero, so the limit when z_{0} = 8+i does not exist.