There exist rational functions with graph having any specified number (0, a positive integer, or infinite) of intersections with the horizontal asymptote; when that number is a positive integer, the behavior at each point of intersection can be arranged to be “crossing” or tangential, as desired. A characterization is given for the rational functions with graph intersecting the horizontal asymptote at finitely many specified x-values and prescribed “crossing” or tangential behavior at each point of intersection. A rational function f, with graph G, has a horizontal asymptote that has infinitely many points of intersection with G if and only if fis a constant function. If fis a nonconstant rational function with a horizontal asymptote, then outside some closed interval of real numbers, fis strictly monotonic on each of the remaining open intervals. Analogous results are given for slant asymptotes. Many examples are given to motivate and illustrate the results. This note could serve as enrichment material in courses on precalculus or calculus.

rational function, horizontal asymptote, point of intersection, polynomial, degree, root, domain, limit, Mean Value Theorem, Intermediate Value Theorem, slant asymptote.