How To Find Horizontal Asymptote Of Exponential Function

In this explainer, we volition learn how to discover the horizontal and vertical asymptotes of a function.

Before we look explicitly at how to find an asymptote of a rational role, permit us recall what an asymptote is.

Definition: Asymptote

An asymptote is a line that approaches a given bend arbitrarily closely. This is illustrated by the graph of .

Here, the asymptotes are the lines and .

In order to place vertical asymptotes of a function, we need to identify any input that does non accept a divers output, and, besides, horizontal asymptotes can often be identified by considering outputs that cannot be reached from whatever input in the function's domain. In item, for rational functions, knowing the domain and range volition aid us to identify the asymptote and vice versa. Therefore, if we are looking to identify the asymptotes of a rational function, we take a very similar arroyo to how we identify the domain and range of a rational function.

If we consider the function

nosotros need to find if it has any undefined input and, as, any values that do not exist in its range. We know that a rational function is undefined when its denominator is zero. Here, is cypher when

This, therefore, is the equation of an asymptote. Are there whatever others? Well, if we consider all the possible outputs of the role and consider what happens as the inputs get progressively large, we tin see that the outputs of the function go closer and closer to zero but can never actually become there. Therefore, we have another asymptote at

By identifying the asymptotes of a rational function, we can easily identify the domain and range. The function has a domain, which is all of the existent numbers except 3, and a range, which is all of the real numbers except 0.

Permit us now await at a couple of examples.

Case 1: Identifying the Asymptotes of Rational Functions

Make up one's mind the vertical and horizontal asymptotes of the role .

Answer

To find the vertical asymptotes of the function, we need to determine if there is any input that results in an undefined output. The function contains two rational expressions that are undefined when their denominators are goose egg. The expression is undefined when , and the expression , over again, is undefined when . Therefore, we have a vertical asymptote with the equation .

To notice the horizontal asymptotes, nosotros need to find if there are any values that do not exist in the range of the role. If nosotros look at the outputs of the office equally the inputs go progressively big, we can run across that the expressions and get closer and closer to null and the function gets closer and closer to merely never really accomplish this value. Thus, we have a horizontal asymptote with the equation .

Before we look at the next example, it is worth noting that a hyperbola is a type of rational role with 2 asymptotes.

Case 2: Identifying the Asymptotes of Rational Functions

What are the ii asymptotes of the hyperbola ?

Answer

To find the vertical asymptotes of the part, we demand to make up one's mind if there is any input that results in an undefined output. The function contains a rational expression that is undefined when its denominator is nada. The expression is undefined when and, therefore, has an asymptote with the equation .

To notice the horizontal asymptotes, we need to notice if in that location are any values that practice not exist in the range of the function. If nosotros look at the outputs of the function as the inputs get progressively large, we can see that the expression gets closer and closer to zero and the value of the function gets closer and closer to but will never actually attain this value. Thus, nosotros accept a horizontal asymptote with equation .

As previously mentioned, if nosotros place the asymptotes of a rational function, nosotros tin utilise this information to hands find the domain and range of the function, and, equally, we can use the information about the asymptotes to aid us sketch or identify the graph of the function. Let u.s.a. now look at examples of this.

Example 3: Using the Asymptotes of a Rational Role to Find Its Domain and Range

Determine the domain and the range of the office in .

Reply

In this question, we are fortunate to have been given the graph of the rational function. This allows the states to hands identify the equations of the asymptotes: nosotros can see that the equation of the vertical asymptote is , and that the equation of the horizontal asymptote is . Using this data, we tin state that the domain of the function is and that the range of the office is .

Suppose we were not given the graph of the office. Nosotros could identify the asymptotes by looking for whatsoever input that results in an undefined output and looking for any output that cannot be achieved regardless of the input. Here, we can come across that we have an undefined output for an input of zero every bit is undefined. Hence, we have an asymptote with the equation . Nosotros can besides encounter that as gets progressively large the outputs of the function become closer and closer to but can never accept this value. Therefore, we also accept an asymptote with the equation .

Case 4: Finding the Asymptotes of a Function in order to Identify Its Graph

Which of the following graphs represents ?

Respond

In this question, we can identify the graph of the rational function by determining the position of its asymptotes. If we look at the equation of the function, nosotros can identify any vertical asymptote by identifying any input that results in an undefined output. The function is undefined when the denominator is equal to zero, that is, when . Therefore, nosotros accept an asymptote with the equation .

To find the horizontal asymptotes, we need to find if there are any values that do not exist in the range of the office. As the inputs get progressively large, we can see that the function gets closer and closer to nil, simply it volition never actually accomplish this value. Thus, we take a horizontal asymptote with the equation .

Using this information, nosotros tin can run into that the correct graph is (c).

Example v: Identifying the Domain of a Rational Function

Find the domain of the part .

Answer

This is a particularly interesting question as it is non immediately obvious how the graph matches up with the function. It is very like shooting fish in a barrel with a question like this to mistakenly state the domain and range and equally make assumptions almost the nature of the function, including whether information technology has whatsoever asymptotes.

If we gene the numerator and denominator of the function, we get that

At this point, nosotros can see that the part is not defined at two points: and . If yous input either of these values, yous will go , which is not defined. Using what we know near rational functions, it would be fair to assume that the function, therefore, has asymptotes at these two points. This, nonetheless, is not the case. Providing does not equal 0 or , then nosotros can simplify our function as follows:

which simplifies to

This shows usa that for all values of , excluding 0 and , the function simplifies to a line. At this point, we can determine that the domain of the part is .

The mutual error here is to simplify the function first and so state that the domain is the whole existent numbers, but this is not the example.

Notice that in this example, the domain of the part does non tell us about the asymptotes of the function. In fact, the function has no vertical or horizontal asymptotes. This will be the case where we take rational functions divers by an expression that tin be simplified past canceling common factors in the numerator and denominator.

Primal Points

To detect the vertical asymptotes of the function, nosotros demand to identify any point that would lead to a denominator of zero, but exist careful if the function simplifies—as with the terminal example.
To find the horizontal asymptotes of a rational part, we need to identify any value that the part cannot accept. It can often be helpful to look at the limits of the part to aid y'all in this process.
We tin can use the asymptotes to help united states identify the range and domain of the function.
We tin use the asymptotes to help us sketch or place the graph of the function.