Behavior at infinity. 
Classify the behavior at $\infty$ for $$f(z)=\frac{\sin z}{z^2},\,g(z)=\frac{1}{\sin z},\,h(z)=\exp\left(\tan\frac{1}{z}\right).$$

So I just considered $f(1/z),g(1/z),h(1/z)$ at $z=0$. For $f$ I get $z=0$ as an essential singularity, but I'm not sure about $g$ and $h$. Also I'm stuck on evaluating $\lim_{z\to0}f(1/z),g(1/z),h(1/z)$ (so I can classify the behaviors at infinity). If $x\in\mathbb{R}$ then $\lim_{x\to0}f(1/x)=0$.

Assuming $z \in \mathbb{R}$, \sin{z} is bounded $\forall z \in \mathbb{R}$.  Therefore 
$$\lim_{z \rightarrow \infty} \frac{\sin{z}}{z^2} = 0$$
$1/\sin{z}$, on the other hand, varies between $1$ and $\infty$ $\forall z \in \mathbb{R}$.  Therefore this limit is undefined.
Because $\exp{(\tan{z})}$ is uniformly continuous, its limit as $z\rightarrow 0$ is $\exp{(0)} = 1$.
