Only an essential singularity lets $f(z_n)$ have no limit, even $\infty$

For which of the following functions $f$, there is a sequence $\{z_n\}$ in $\mathbb{C}-\{1\}$ such that $\lim z_n=1$, but $f(z_n)$ has no limit in $\mathbb{C}\cup\{\infty\}$?