Horizontal Asymptotes

In mathematics, a horizontal asymptote is a straight horizontal line that a graph approaches but never touches as the input values increase or decrease. Horizontal asymptotes are used to describe the behavior of functions as the input values become very large or very small.

Finding Horizontal Asymptotes

To find the horizontal asymptote of a function, you can use the following steps:

1. Find the degree of the numerator and the degree of the denominator of the function.
2. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
3. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.
4. If the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote.

Examples

Let's look at some examples to understand how to find horizontal asymptotes:

Example 1: Find the horizontal asymptote of the function f(x) = (3x² + 2) / (x - 1).

Solution:
Step 1: The degree of the numerator is 2 and the degree of the denominator is 1.
Step 2: Since the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote.

Example 2: Find the horizontal asymptote of the function g(x) = (2x³ - 5x + 1) / (x³ + 4x - 2).

Solution:
Step 1: The degree of the numerator is 3 and the degree of the denominator is also 3.
Step 2: Divide the leading coefficients: 2 / 1 = 2.
So, the horizontal asymptote is y = 2.

Study Guide

When studying horizontal asymptotes, make sure to focus on the following key points:

• Understanding the concept of horizontal asymptotes and their significance in describing the behavior of functions.
• Learning the steps to find horizontal asymptotes for rational functions.
• Practicing various examples to strengthen your understanding of finding horizontal asymptotes.

Remember to review and practice regularly to master the concept of horizontal asymptotes!