Eventually bounded rational functions

In real variable analysis, it is an easy fact to show a ratio of polynomials of equal degree is eventually bounded by using L'Hopital. This fact seems to be extended to rational functions of a complex variable by virtue of being "obvious", but I cannot see an obvious way to go about it! Specifically, I cannot see a way to minimize an arbitrary polynomial on a circle in the complex plane (e.g. x^n+1 is always greater than R^n-1 be considering the geometry of the situation).

How would one show this seemingly obvious fact?