Let $\displaystyle f$ be holomorphic on $\displaystyle U$\$\displaystyle \{P\}$ where $\displaystyle P \in U$ and $\displaystyle U$ is open. If $\displaystyle f$ has an essential singularity at $\displaystyle P$, then what type of singularity does $\displaystyle 1/f$ have at $\displaystyle P$? what about when $\displaystyle f$ has a removable singularity or a pole at $\displaystyle P$?