Consider the function $$f(z)=\frac{\tan(z)}{z}, \quad z \in \mathbb{C}$$
I want to find it's singular points, determine which are removable and classify the smallest positive non-removable singular point.
I know that $f$ has a singular point at $z=0$, which is removable since $\lim_{z \to 0}f(z) \neq0$. How do I find any others and then classify the smallest non removable one though?
HINT:
Note that $\tan(z)=\frac{\sin(z)}{\cos(z)}$ and $\cos(z)=0$ for $z=(n+1/2)\pi$ where $n$ is any integer.
Then, note that $\lim_{z\to (n-1/2)\pi}\frac{z-(n-1/2)\pi}{\cos(z)}=(-1)^n$