Backstory. I was trying to come up with a function $f(z)$ that would have an essential singularity in $z = 0$ and would be limited by the following function with non-integer power (for some $\alpha$ and $A$): \begin{equation} |f(z)|\leq\frac{A}{|z|^\alpha},\ \alpha\in(0,1),\ 0<|z|<1. \end{equation}
The question. Is there a function $f(z):\ O_r(\infty) \to \mathbb{C}$ such that $f(z)$ is holomorphic in some viscinity $O_r(\infty)\subset\mathbb{C}$ of the infinity and $$\lim_{z\to\infty}\frac{f(z)}{z}=0$$ whilst $$\nexists\lim_{z\to\infty} f(z)$$ ?
Referring to the backstory, $f(\frac{1}{z})$ should then satisfy the original inequality.
Such a function does not exist. $|f(z)|\leq\frac{A}{|z|^\alpha}$ with $ \alpha\in(0,1)$ implies that $$ \lim_{z \to 0} z f(z) = 0 $$ and then Riemann's theorem implies that $f$ has a removable singularity at $z=0$.