Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches as the input increases or decreases without bound. It describes the long-term behavior of a function.

Consider two simplified models of drug concentration in the bloodstream, one for a single dose and another for continuous supply.

After a single dose, the concentration can be modeled by a rational function where the numerator has a lower degree than the denominator.

As time increases, the denominator grows, so the concentration approaches zero. The graph approaches the horizontal line y equals zero, forming a horizontal asymptote.

With a continuous supply, we can use a rational function in which the numerator and denominator have the same degree.

In this model, the concentration approaches a constant value. Since both degrees are equal, the leading coefficients of 6t and t are 6 and 1. Their ratio is 6, giving the horizontal asymptote y equals 6.

These models show how degrees determine horizontal asymptotes for rational functions. A lower degree in the numerator gives y = 0, equal degrees use the leading-coefficient ratio, and a higher degree in the numerator gives no horizontal asymptote.