Horizontal Asymptotes: Definition & Rules

Given a particular function, there is actually a 2-step procedure we can use to find the horizontal asymptote. Learn what that is in this lesson along with the rules that horizontal asymptotes follow.

Definitions

Before getting into the definition of a horizontal asymptote, let’s first go over what a function is. A function is an equation that tells you how two things relate. Usually, functions tell you how y is related to x. Functions are often graphed to provide a visual.

A horizontal asymptote is a horizontal line that tells you how the function will behave at the very edges of a graph. A horizontal asymptote is not sacred ground, however. The function can touch and even cross over the asymptote.

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Horizontal asymptotes exist for functions where both the numerator and denominator are polynomials. These functions are called rational expressions. Let’s look at one to see what a horizontal asymptote looks like.

horizontal asymptote

So, our function is a fraction of two polynomials. Our horizontal asymptote is y = 0. Look at how the function’s graph gets closer and closer to that line as it approaches the ends of the graph. We can plot some points to see how the function behaves at the very far ends.

x y
-10,000 -0.0004
-1000 -0.004
-100 -0.04
-10 -0.4
-1 -4
1 4
10 0.4
100 0.04
1000 0.004
10,000 0.0004

Do you see how the function gets closer and closer to the line y = 0 at the very far edges? This is how a function behaves around its horizontal asymptote if it has one. Not all rational expressions have horizontal asymptotes. Let’s talk about the rules of horizontal asymptotes now to see in what cases a horizontal asymptote will exist and how it will behave.

Horizontal Asymptotes Rules

There are three rules that horizontal asymptotes follow depending on the degree of the polynomials involved in the rational expression. Before we begin, let’s define our function like this:

horizontal asymptote

Our function has a polynomial of degree n on top and a polynomial of degree m on the bottom. Our horizontal asymptote rules are based on these degrees.

  1. When n is less than m, the horizontal asymptote is y = 0 or the x-axis.
  2. When n is equal to m, then the horizontal asymptote is equal to y = a/b.
  3. When n is greater than m, there is no horizontal asymptote.

The degrees of the polynomials in the function determine whether there is a horizontal asymptote and where it will be. Let’s see how we can use these rules to figure out horizontal asymptotes.

Finding a Horizontal Asymptote: Step 1

Let’s find the horizontal asymptote to this function:

horizontal asymptote

Our first step is to make sure our function is written in standard form in both the numerator and denominator. Standard form tells us to write our largest exponent first followed by the next largest all the way to the smallest. Looking at our function, it looks like it already is in standard form. We are good to go here. We can move on to the second step.

Finding a Horizontal Asymptote: step 2

Next, we are going to rewrite the function with only the first terms in both the numerator and denominator. We are removing all the other information after the term with the largest exponent. Our function ends up looking like this:

horizontal asymptote

Now, we can use the rules to find our horizontal asymptote. I first need to compare the degree of the numerator to the degree of the denominator. I see that they are the same, so that means my horizontal asymptote is the fraction of the coefficients involved, which is y = 3/5. That was easy.

Lesson Summary

To recap, a horizontal asymptote tells you how the function will behave at the very edges of the graph going to the far left and the far right. The three rules that horizontal asymptotes follow are based on the degree of the numerator, n, and the degree of the denominator, m.

  1. If n ; m, the horizontal asymptote is y = 0.
  2. If n = m, the horizontal asymptote is y = a/b.
  3. If n ; m, there is no horizontal asymptote.

Learning Outcomes

After viewing this lesson, you should be able to:

  • Describe the 2-step procedure used to find a horizontal asymptote
  • Examine the rules of horizontal asymptotes in terms of ‘greater than,’ ‘less than,’ or ‘equal to’
x

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