This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…

Three proofs are given for the fact that the derivative of an everywhere-positive non-constant real polynomial function must change sign. This self-contained note could find classroom use in courses on calculus or abstract algebra.