How to Find Horizontal Asymptotes in 3 Easy Steps

Step-by-Step Guide: How to Find the Horizontal Asymptote of a Function

During your study of algebra, you will eventually learn how to analyze and understand the behavior functions when they are represented graphically on the coordinate plane. The graph of a function gives you a visual portrayal of how the function behaves—including the location of any potential horizontal asymptotes.

This free step-by-step guide on How to Find Horizontal Asymptotes of a Function will demonstrate and explain everything you need to know about horizontal asymptotes and how to find the horizontal asymptote of any function (assuming that the function does indeed have a horizontal asymptote).

The following sections will be covered in this guide. While we recommend that you work through each section in order, you can use the quick-links below to jump to a particular section":

• What is a Horizontal Asymptote?

• How to Find Horizontal Asymptotes Example #1

• How to Find Horizontal Asymptotes Example #2

• How to Find Horizontal Asymptotes Example #3

Let’s start off by learning more about horizontal asymptotes and what they look like.

What is a Horizontal Asymptote?

Before you learn how to find horizontal asymptotes, it is important for you to understand some key foundational concepts.

In algebra, a function is an equation of the form y=f(x) that represents the relationship between two sets. A relation can be defined as a function when there is only one output value for each input value (i.e. the x-values do not repeat). Functions can be visually represented on a graph (i.e. a coordinate plane).

A horizontal asymptote of a function is a horizontal line (↔) that tells you the behavior of a function as it approaches the edges of a graph. We can also say that horizontal asymptotes allow us to identify the “end behavior” of a function.

It is also important to note that horizontal asymptotes occur when functions are rational expressions, meaning that the function is a quotient of two polynomials (i.e. the function is a fraction where both the numerator and the denominator are polynomials.

For example, consider the following function:

• y = (3x²)/x³

Our function, y = (3x²)/x³, is the quotient of two polynomials (i.e. it is a rational expression) and its corresponding graph is shown in Figure 02 above.

Since our function is a fraction, we know that we can not have zero in the denominator (otherwise the function would be undefined at this point). So, the graph of the function, as you can see in Figure 02, does something interesting at zero.

In this example, we have a horizontal asymptote at y=0.

Notice how the graph of the function y = (3x²)/x³ gets closer and closer to the line y=0, without ever touching it, as it approaches the ends of the graph (horizontally), as shown in Figure 03 below.

If we take a look at a few points on the graph of y = (3x²)/x³ and the corresponding table, as shown in Figure 04 below, we can see that as the graph moves to the left of the coordinate plane, the y-values get closer and closer to zero. And, we can also see that as the graph moves to the right of the coordinate plane, the y-values get closer and closer to zero.

This type of observation is what we call determining the “end behavior” and it helps us to understand why the graph has a horizontal asymptote at y=0.

Now that you understand what a horizontal asymptote is and what it looks like, you are ready to work through a few step-by-step examples of how to find a horizontal asymptote.

Before moving forward, here are a few quick key points you should be familiar with:

• Not all functions that are rational expressions have a horizontal asymptote.

• It is possible for a function to have 0, 1, or 2 asymptotes (i.e. a function can have a maximum of 2 asymptotes).

• When horizontal asymptotes are shown on a graph, they are typically drawn using a dashed line, which is what you will see in this guide.

• The horizontal asymptote of a function is not a part of the function, and it is not a requirement to include the horizontal asymptote of a function when you graph it on the coordinate plane.

• A horizontal asymptote can be thought of as an imaginary dashed line on the coordinate plane that helps you to visual a “gap” in a graph.

How to Find Horizontal Asymptotes Example #1

Now you are ready to learn how to find a horizontal asymptote using the following three steps:

• Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

• Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

• Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Now, let’s go ahead and apply these three steps to our first practice problem.

Example #1: Find the horizontal asymptote of the function f(x)=4x/(x+4)

For this first example, we can start by taking a look at the graph in Figure 05 above. By looking at the graph, we can see that the function f(x)=4x/(x+4) has one horizontal asymptote at y=4.

Let’s now apply our three steps to see if confirm the at the function has a horizontal asymptote at y=4.

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

• lim x→∞ f(x) = lim x→∞ 4x/(x+4)

• = 4 lim x→∞ [1/(1+(4/x))]

• = 4 (lim x→∞ (1)) / (lim x→∞ (1+(4/x))

• = 4 ( (1)/(1))

• =4

After completing the first step, we can see that, as the function has a horizontal asymptote at y=4 as the function approaches positive ∞ on the graph.

Next, let’s confirm that the function has the same behavior as it approaches negative ∞.

Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

• lim x→-∞ f(x) = lim x→∞ 4x/(x+4)

• = 4 lim x→-∞ [1/(1+(4/x))]

• = 4 (lim x→-∞ (1)) / (lim x→∞ (1+(4/x))

• = 4 ( (1)/(1))

• =4

Completing the second step gives us the same result as the first step (i.e. there is a horizontal asymptote at y=4).

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Finally, based on our results from steps one and two, we can conclude that the function has one horizontal asymptote at y=4, as shown on the graph in Figure 06 below.

Final Answer: The function has one horizontal asymptote at y=4.

How to Find Horizontal Asymptotes Example #2

Find the horizontal asymptote of the function f(x)=3ˣ+5

For this next example, we want to see if the exponential function f(x)=3ˣ+5 has any horizontal asymptotes.

We can solve this problem the same as we did the first example by using our three steps as follows:

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

• lim x→∞ f(x) = lim x→∞ 3ˣ+5

• =lim x→∞ (3ˣ) + lim x→∞ (5)

• lim x→∞ (3ˣ) = ∞

• lim x→∞ (5) = 5

• = ∞ + 5 = ∞

Notice that our result for the first step is that, as the function approaches positive ∞, the limit is ∞. And, since ∞ is not a real number, we can not yet determine whether or not this function will have any horizontal asymptotes. But we can not be completely sure until we complete the second step.

Step Two: Determine lim x→-∞ f(x. In other words, find the limit for the function as x approaches negative ∞.

• lim x→-∞ f(x) = lim x→-∞ 3ˣ+5

• =lim x→-∞ (3ˣ) + lim x→-∞ (5)

• lim x→-∞ (3ˣ) = 0

• lim x→-∞ (5) = 5

• = 0 + 5 = 5

Our result from step number two results in a real number, so we can conclude that the function does have a horizontal asymptote at y=5.

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Again, even though Step One did not result in a real number limit, Step Two did, so we can conclude that:

Final Answer: The function has one horizontal asymptote at y=5.

We can confirm the location of a horizontal asymptote at y=5 for the exponential function f(x)=3ˣ+5 by looking at the completed graph in Figure 07 below.

How to Find Horizontal Asymptotes Example #3

Find the horizontal asymptote of the function f(x)=(3x^2+x)/(x+2)

For this third and final example, we have to see if the rational function f(x)=(3x^2+x)/(x+2) has one, two, or zero horizontal asymptotes.

We can determine whether or not the function has any horizontal asymptotes by following our three steps as follows:

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

• lim x→∞ f(x) = lim x→∞ (3x^2+x)/(x+2)

• =lim x→∞ (3x+1)/(1+(2/x))

• = (lim x→∞ (3x+1)) / (lim x→∞(1+(2/x)))

• lim x→∞ (3x+1) = ∞

• lim x→∞(1+(2/x)) = 1

• = ∞ / 1 = ∞

Since our result is not a real number, Step One does not help us to determine the possible existence of a horizontal asymptote.

Step Two: Determine lim x→-∞ f(x. In other words, find the limit for the function as x approaches negative ∞.

• lim x→-∞ f(x) = lim x→-∞ (3x^2+x)/(x+2)

• =lim x→-∞ (3x+1)/(1+(2/x))

• = (lim x→-∞ (3x+1)) / (lim x→-∞(1+(2/x)))

• lim x→-∞ (3x+1) = -∞

• lim x→-∞(1+(2/x)) = 1

• = -∞ / 1 = -∞

Again, our result is not a real number so we can not determine the location of a horizontal asymptote.

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Since neither Step One nor Step Two resulted in a real number value for k, we can conclude that the function does not have any horizontal asymptotes.

Final Answer: The function does not have any horizontal asymptotes.

The graph of the function f(x)=(3x^2+x)/(x+2) is shown in Figure 08 below. Do you notice how there is no horizontal asymptote?

Conclusion: How to Find Horizontal Asymptotes

A horizontal asymptote of a function is an imaginary horizontal line (↔) that helps you to identify the “end behavior” of the function as it approaches the edges of a graph.

Not every function has a horizontal asymptote. Functions can have 0, 1, or 2 horizontal asymptotes.

If a function does have any horizontal asymptotes, they will be displayed as a dashed line. A horizontal asymptote is an imaginary line that is not a part of the function, and it is not a requirement to include the horizontal asymptote of a function when you graph it on the coordinate plane.

You can determine whether or not any function has horizontal asymptotes by following these three simple steps:

• Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

• Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

• Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Keep Learning:

How to Find the Vertex of a Parabola in 3 Easy Steps

Learn how to find the coordinates of the vertex point of any parabola with this free step-by-step guide.