8.3: Long-Run Behavior of Rational Functions

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Objectives:

• Identify rational functions and find possible horizontal asymptotes
• Give long-run behavior of rational functions (find a power function that describes the behavior for large values of $x$)

Important Items

Definitions:

rational function

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Lesson Guide

Warm-Up

Have students do Problem 1.

Identify rational functions and find possible horizontal asymptotes

A rational function $r(x)$ is a function that can be written in the form $r(x)=\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x)\not=0$.

Have students do Problems 2 and 3.

Use Problem 3 to discuss what the long-run behavior tells us about horizontal asymptotes. Refer back to the definition of a horizontal asymptote from \S4.4.

Give long-run behavior of rational functions (find a power function that describes the behavior for large values of $x$)

Give examples of rational functions that illustrate different types of long-run behavior (i.e., $\pm nfty$ vs. horizontal asymptote).

• Example:

• Example:

Have students do Problems 4 and 5.

You may need to review limit notation for Problem 5.

Discuss Problem 6 as a whole class if time permits.

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