Help for essential singularity problem

[Vlad]

Hello, can anyone help me out here?
If you have a function f(z) in U, and b in U, such that b is an isolated essentially singular point for f(z) in U, what type of singularity can
g(z) = 1/f(z) have?

[HallsofIvy]

Try thinking about it the other way around. Suppose f(z) is analytic at q but f(q)= 0. What kind of singularity can 1/f(z) have at q?

Suppose f(z) has a pole at q. What kind of singularity can 1/f(z) have at q?