Prove two points that might be essential singularity

Question

$f(z)=\frac{ze^{\frac{1}{(1-z)^2}}}{sin{\pi z}}$ I can find that $z=\mathbb Z$$-${0,1} are poles of order 1. And $z=0$ is a removable singularity. But how to judge the behavior of $f(z)$ at $z=1$ and $z=\infty$ ? I think they might be essential singularity due to the $e^z$ and $\sin{z}$ term. But how to prove them?

Answer 1

What is $\lim_{z\to1}f(z)$? (And now what's the definition of "pole"?)

Answer 2

At $z=1$ the $z/\sin(\pi z)$ part has a simple pole but the exponential part has a non-terminating Laurent expansion, so it is an essential singularity. As $z\rightarrow \infty$ there are more than one limit point on the Riemann sphere so it is not a finite order pole.