Let $f$ be a holomorphic on $\Bbb{C}\setminus\{0\}$, so that $z=0$ is a pole of order $m$ of $f$, and so that the limit $\lim_{z\to\infty} f(z)$ exists (finite or infinite). Show that $f$ is in the form $$f(z)=\frac{P(z)}{z^m},$$ for a polynomial $P(z)$, so that $P(0)\ne 0$.
My attempt: consider $g(z)=f(\frac{1}{z})$. Then $\lim_{z\to 0} g(z)$ exists (finite or infinite). I am trying to show that $g(z)=z^m\psi(z)$ for some meromorphic $\psi$. How to continue?