Thread: 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?

Originally Posted by siclar

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.

You don't need L'Hopital's rule for that: if you divide denominator and numerator by the unknown to the power of the common degree, you see that for the rational function has a finite limit. And because it has a limit, it is "eventually bounded".