In Rudin's book I've found this exercise:
Let $f$ be a continuous complex function defined in the complex plane. Suppose there is a positive integer $n$ and a complex number different from $0$ such that: $$\lim_{z\to \infty} z^{-n}f(z)=c$$Prove that $f(z)=0$ for some complex number $z$.
There is a hint that suggest to use the winding number. Is there a simple proof to this problem?