What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C$? What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C^*$? I need it to determine the type of the singularity at $z = 0$. 

It is undefined. Consider the function $g(z)=\frac{\sin{z}}{z}$. This can be made an entire function by setting $g(0)=0$. But then you are looking for $\lim_{z\to\infty} g(z)$. Any non-polynomial entire function must have an essential singularity at $\infty$, so this limit does not exist.
Alternatively:
For $x$ real, $\lim_{x\to\infty} g(x)$ is clearly zero, since $|\sin x|\leq 1$.  
But when $z=ix$ is imaginary, then $g(z)=\frac{e^x-e^{-x}}{2x}$, which has infinite limit as $x\to\infty$.
