In earlier lessons, we saw rational functions whose end behavior could be described by a horizontal asymptote. For example, we can rewrite functions like as to see more clearly that as gets larger and larger in either the positive or negative direction, the value of gets closer and closer to 0, which means the value of gets closer to 1. We can use similar thinking to understand rational functions that do not have horizontal asymptotes.

For example, consider . Using division, the expression can be rewritten as . As gets larger and larger in either the positive or negative direction, the value of the term gets closer and closer to 0, which means the value of  gets closer to the value of . This means that the end behavior of can be described by the line . Here is a graph of , the diagonal line , and the vertical asymptote of the function at :

Graph of a rational function f of x with no horizontal asymptotes, two dashed intersecting lines on coordinate plane. Each axis from -20 to 35, by 10’s. Vertical asymptote of the function at x = 3 and the line y = x + 7 intersect at (3 comma 10).