Let be given the functions: $\ \forall z \in \mathbb{C}\displaystyle{f_{1}(z) = \frac{1}{(z-1)^{3}},(z\neq 1); f_{2}= \cos\left(\frac{1}{z}\right)}; (z\neq 0)$ and let $U$ be an open disc with radius $R$ around $z_{0}=1$ for $f_{1}$ and $z_{0}=0$ for $f_{2}$.
Onto which set $V\subset \mathbb{C}$ is the unit disc mapped? How many times do $f_{1}, f_{2}$ take each value $w\in V$ on $U\backslash \{z_{0}\}$?
$f_1$ has a pole of order $3$ at $z_{0}$ , and $\cos(1/z)$ has an essential singularity at $z_{0}$. $f_{1}$ takes each value $w$ $3$ times and $f_{2}$ once. Since it is an essential singularity in $f_2$, that means the map of the disc will be dense and thus onto whole $\mathbb{C}$. For $f_{2}$ since it has a pole of order $3$, the map will not be dense in $\mathbb{C}$.
Can one say onto which set exactly the disc with radius $R$ is mapped onto? How?