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# How To Find Horizontal Asymptotes

6 min readdecember 13, 2021

Jane Zhao

### Precalculus📶

Bookmarked 1k • 4 resources
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## Rules and Examples for Finding Horizontal Asymptotes

### What is an Asymptote?

Asymptotes are an important topic that you’ll see throughout math: from Algebra II all the way to AP Calculus. As you get more and more advanced, the applications of asymptotes will naturally get more complicated. For now, let’s stick with the basics!
First, what exactly is an asymptote? Good question! An asymptote is a line that a function approaches as it heads to infinity without actually intersecting, ever.
Pro tip 🤓 : To help us visualize this concept, think of the relationship between an asymptote and its function as oil and water. They don’t mix, ever. ❌
Also, asymptotes are most commonly found in rational functions (at least for high school mathematics). These are functions that can be written as a ratio of two polynomial functions. It’s also important to note that the polynomial in the denominator cannot equal zero! This is what causes a vertical asymptote, which we talk about more here.

### Why are Asymptotes Important?

Asymptotes can tell us a lot about a function’s end behavior and even help us sketch its graph. It’s like the curb on the side of the street, guiding the function (car) and shaping the way it goes. 🚗

### What is a Horizontal Asymptote?

Primarily, there’s two different types of asymptotes: horizontal and vertical. In this guide, we’ll be focusing on horizontal asymptotes. Make sure to go check out the guide on vertical asymptotes after you read this one!
A horizontal asymptote, like the name suggests, is horizontal. Surprising, right? 😱 Parallel to the x-axis, horizontal asymptotes help describe the behavior of the graph depending on the input of the function, or the x-value.
As x goes towards infinity or negative infinity, the function approaches a horizontal asymptote (or two, depending on the specific function).

Example Function

The exponential function, f(x) = ex, has a horizontal asymptote at y = 0. On a zoomed out graph, like the one below, it really looks like the two functions are touching! However, if we zoom in really close, we can see that they actually never intersect. 🔎

Image from Desmos.

### How to Find a Horizontal Asymptote of a Function

To find the horizontal asymptote(s) of a function, make sure to rewrite the function in standard form if it isn’t given to you like that already. It’ll make everything easier in the long run! Remember that standard form just means that everything is multiplied out and the terms go with the highest degree first.
Next, there are three different possibilities.
1. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
2. If the degree of the numerator is equal to the degree of the denominator, then the horizontal asymptote is the ratio of the leading coefficients of the polynomial functions.
3. If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is equal to the x-axis, or y = 0.
Let’s put this information into a table to visualize it better! 🤓

Horizontal Asymptotes of Rational Functions

Let deg N(x) = the degree of the numerator and deg D(x) = the degree of the denominator.
 deg N(x) > deg D(x) deg N(x) = deg D(x) deg N(x) < deg D(x) y = DNE (does not exist)(There is no horizontal asymptote.) y = leading coefficient of N(x)leading coefficient of D(x)
Alright, so what are the steps we need to take to find a horizontal asymptote?
1. Check that the rational function is in standard form.
2. Identify the degree of the numerator.
3. Identify the degree of the denominator.
4. Determine the relationship between the degree of the numerator and the degree of the denominator.
5. Write the horizontal asymptote as an equation in the form, y = x, where x is the value the graph approaches.
Let’s apply what we just learned to a couple functions to see if we really understood!

Practice, Practice, Practice!!

Example 1:
f(x) = x^2-18x+81x^5-6x^3+7x^2-29
Okay, so we’re given the above function and are asked to determine whether or not it has horizontal asymptotes and to identify them if it does.
First, we check to see that the two polynomials are written in descending degrees. Looks like both are good to go, so we move on to the next step!
Let’s identify the degree of the two polynomials. For the numerator, the highest degree is 2. For the denominator, the highest degree is 5.
Now, we compare the two values. Two is less than five, so we have the following situation, where deg N(x) < deg D(x). This means that we have a horizontal asymptote of… you got it! y = 0.
We can double-check our answer by graphing the function on a calculator and seeing where the graph approaches.

Image from Desmos.

As the graph shows, the function approaches the x-axis, but never actually intersects it. There seems to be some vertical asymptotes as well, but we’ll worry about those later!
Example 2:
Again, we’re presented with a function and asked to find the horizontal asymptotes. The polynomials both appear to be written in standard form, so we’re good on that! The degree for the numerator is 3, and the degree for the denominator is 3.
Great! We’ve identified the degree of the function and just need to look at the relationship between the two values. 3 is equal to 3, so we have the situation where deg N(x) = deg D(x). Our next step is to take the leading coefficient of the first terms and rewrite it in a ratio.
The first term of the numerator is x^3, which has a leading coefficient of 1. Not 0! Remember that there’s always a 1 in front of terms that stand alone -- we just don’t write it for simplicity’s sake!
The first term of the denominator is -6x^3. Looking at the coefficient, we see that it is -6.
Now, we write these two values into a fraction and get -1/6 as our answer, Thus, the function f(x) has a horizontal asymptote at y =  -1/6.

Image from Desmos.

Example 3:
Let’s do one last problem together! Y’all know the drill now 😌. Now, we check if the function’s written in standard form. This one’s not! We have to rewrite the equation with the terms going in descending degree order. The highest degree of the numerator is 5, so we rewrite the polynomial and get -12x^5+x^3+7x-45. Next, we do the same to the denominator. The highest degree is 5, which is already in the first position. The second highest degree is 3, not 2, so we rewrite the polynomial and get x^5+23x^3-76x^2+17. The rewritten function is
Now, the degrees of the numerator and denominator are both 5, which means that we have to take the ratio of the leading coefficients. The leading coefficient of the x^5 term in the numerator is -12, while the leading coefficient of the x5 term in the denominator is 1. Therefore, the horizontal asymptote of f(x) is y = -12.

Image from Desmos.

Alright, now that we’ve gotten some practice problems in, you should be good to go! If you’re still unsure on the concept, reread the content and try out some simpler problems with fewer terms and work your way up to the more complex functions. Make sure to go check out this article on how to find vertical asymptotes as well! You got this! 💪
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