Horizontal Asymptotes: Definition & Rules

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Linear Inequality: Solving, Graphing & Problems

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:02 Definitions
  • 1:24 Horizontal Asymptote Rules
  • 2:20 Step 1
  • 2:44 Step 2
  • 3:18 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Expert Contributor
Alfred Mulzet

Dr. Alfred Kenric Mulzet received his Ph.D. in Applied Mathematics from Virginia Tech. He currently teaches at Florida State College in Jacksonville.

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.


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.

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

To unlock this lesson you must be a Study.com Member.
Create your account

Additional Activities

Horizontal Asymptotes:

We learned that if we have a rational function f(x) = p(x)/q(x), then the horizontal asymptotes of the graph are horizontal lines that the graph approaches, and never touches. But is this always the case?

Exercise 1:

Consider the function f(x) = (2x - 4)/(x + 1). Make a table of values for the function, using the x values ± 10, ± 100, ± 1000. Round your answers to two decimal places. What happens to the y values? Create a graph of the function.


The y values approach 2. Also observe that the numerator and the denominator are of the same degree (degree = 1) and the ratio of the highest terms is 2x/x = 2. This is an analytical way to see the horizontal asymptote y = 2. This is a general feature: If the numerator and denominator are of the same degree, then there will be a horizontal asymptote that can be found by taking the ratio of the highest term of the numerator over the highest term of the denominator.

A graph of the function appears below:

Exercise 2:

Now consider the function f(x) = (x - 2)/(x2 - 9). The degree of the numerator is one, and the degree of the denominator is two. Since the degree of the numerator is less than the degree of the denominator, the line y = 0 is a horizontal asymptote for the graph. Does the graph have an x intercept? Remember that the x intercept is where y = 0. What does this tell you about whether a graph can cross a horizontal asymptote or not?


The graph of this function crosses its horizontal asymptote at x = 2. A graph of this function appears below:

Exercise 3:

Finally, consider the function f(x) = (2x2 + 4)/(x - 2). The degree of the numerator is two, and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, the graph does not have a horizontal asymptote. Here is a graph of the function:

Although this graph does not have a horizontal asymptote, it does have what is known as an oblique, or diagonal, asymptote. Can you see where this is? Imagine drawing a diagonal line through the graph, and see how you can make it almost touch the graph. It is possible to obtain the equation of the oblique asymptote. Simply divide the numerator of the function by the denominator, and throw away the numerator. This gives the equation

y = 2x + 4.

Now here is a graph of the same function, with the oblique asymptote included:

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account