The study of asymptotes constitutes one of the earliest and most significant encounters a high school mathematics student can have with infinity. The study of horizontal asymptotes, in particular, contributes to a student's later understanding of limits and the notion of “arbitrarily close to.” Typically, students are exposed to horizontal asymptotes in the contexts of exponential and rational functions. Real–life applications of exponential functions abound, but real–life applications of rational functions do not. In this article, we present a hands–on activity that allows students to explore horizontal asymptotes of graphs of rational functions visually and mathematically and that has been used successfully with intermediate algebra, precalculus, college algebra, and mathematics methods students.