Finding Horizontal Asymptotes

Pre-Calculus · Axiom Academy

EXAMPLE Finding Horizontal Asymptotes Compare the degrees of the numerator and denominator to locate the horizontal asymptote of a rational function. Degree of top degree of bottom HA is y = 0 . Degree of top degree of bottom no horizontal asymptote. Find the horizontal asymptote of each rational function by comparing the degree of its numerator with the degree of its denominator. Both ends of the graph press toward the dashed line y = 2 — the ratio of the leading coefficients, 4 over 2. Nice work — you located three horizontal asymptotes by reading nothing but the leading terms. Lower top: when the numerator's degree is smaller, the bottom wins and the HA is y = 0 . Equal degrees: the HA is the ratio of the leading coefficients — here . Higher top: the function grows without bound, so there is no horizontal asymptote (it has a slant asymptote instead). You never had to expand or divide — for end behavior, only the highest-power terms matter.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.