How to Find the Horizontal Asymptote: With examples!

Whether you are taking AP calculus or A-level mathematics, finding horizontal asymptotes is among the key concepts to grasp in calculus. The truth is finding horizontal asymptotes is easy if you know the right steps.

So, how to find the horizontal asymptote? To find the horizontal asymptote, determine the degree of the numerator and denominator and then compare them.

• If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0.
• If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is y = a/b, where a is the coefficient of the highest power of x in the numerator, and b is the coefficient of the highest power of x in the denominator.
• If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.

Read on to find out how to find the horizontal asymptote with examples. Check out this article if you wonder How to Find Vertical Asymptotes of Rational Functions.

What is a Horizontal Asymptote?

A horizontal asymptote is a straight line y = b that a curve approaches as the x-values get very large ( ∞) or very small (–∞). In other words, it describes what happens to a function as the input gets infinitely large or small. A horizontal asymptote can be written as y = b, where b is a constant value.

As a given function approaches infinity on the x-axis, the value of y will approach a certain value known as the horizontal asymptote. It is important to note that a function may not cross or touch the horizontal asymptote.

To learn more about horizontal asymptotes, I encourage you to check out Andrews University’s article or watch this video.

Types of Asymptotes

Functions can have three types of asymptotes:

• A horizontal asymptote at y = b, where b is a constant.
• A slant asymptote, a function in the form of y = mx + b.
• A vertical asymptote is a vertical line x = a where the graph approaches positive (∞) or negative (–∞) infinity as the inputs approach a.

How to Find The Horizontal Asymptote

To find the horizontal asymptote of a rational function, you can compare the degrees of the polynomials in the numerator and denominator: 

• If the degree of the numerator is smaller than the degree of the denominator, meaning the horizontal asymptote is y = 0. 
• If the degree of the numerator is bigger than the degree of the denominator, it means that there is no horizontal asymptote. 
• If the degree of the numerator is equal to the degree of the denominator; therefore, the horizontal asymptote is y = ratio of leading coefficients.

For example, let’s say we have the function:

f(x) = (3x² + 4x + 1) / (4x² – 6x + 2)

Here, the degree of the numerator and denominator is 2. To find the horizontal asymptote:

• We compare the leading coefficients of the numerator and the denominator, which are 3/4.
• Therefore, the horizontal asymptote for this function is y = 3/4.

Another example is the function g(x) = (x² + 2)/(x – 1).

Using the degree method, we can see that the degree of the numerator is 2 and the degree of the denominator is 1, meaning the degree of the numerator is bigger than the degree of the denominator; therefore, there is no horizontal asymptote.

If you want to learn how to find the horizontal asymptote, I encourage you to watch this video or the video below or head over to Khan Academy.

What to read next:

• AP Calculus AB vs. AP Calculus AB, which one is harder? (Let’s find out!)
• Best Graphing Calculator for AP Calculus.
• How to Perform Long Division with Polynomials: easy steps by step guide.

Wrapping Up

Finding a horizontal asymptote allows us to understand how a function behaves as x gets very large or very small and can be useful in a variety of applications.

To find a horizontal asymptote, you can use the limit method or the degree method. Whether you are a student or a teacher, understanding how to find the horizontal asymptote is an essential skill in calculus.