Video: Horizontal and Vertical Asymptotes of a Function

In this video, we will learn how to find the horizontal and vertical asymptotes of a function.

14:49

Video Transcript

Horizontal and Vertical Asymptotes of a Function

In this video, we will learn how to find the horizontal and vertical asymptotes of a function by considering certain limits. Let’s start by covering the definition of an asymptote. A horizontal or vertical asymptote of a curve is a straight line such that the distance between the curve and the line approaches zero as either the π‘₯- or 𝑦-coordinate approaches infinity. We can see an example of an asymptote. If we consider the graph of the function 𝑓 of π‘₯ is equal to one over π‘₯. This function has a vertical asymptote at π‘₯ equals zero and a horizontal asymptote at 𝑦 equals zero.

There are various ways in which an asymptote can occur. For vertical asymptotes, one way which the asymptote can occur is if the function tends to positive infinity from the left and negative infinity from the right. Another similar way is if the function approaches negative infinity from the left and positive infinity from the right which is what we saw in the case of 𝑓 of π‘₯ is equal to one over π‘₯. And now, the case is if the function approaches negative infinity from both the left and right, similarly, the curve could approach positive infinity from both the left and right. Alternatively, it may be that only one of the left or right limit is infinite. This could be either the left or right limit tending to either positive or negative infinity.

From this, we can conclude that π‘₯ equals 𝑐 is a vertical asymptote. If as π‘₯ approaches 𝑐 from either left or right, 𝑓 of π‘₯ tends to positive or negative infinity. Now, we can consider horizontal asymptotes. Now, these will occur when the limit as π‘₯ approaches either positive or negative infinity of 𝑓 of π‘₯ is equal to some constant. We say that 𝑦 equals 𝑐 is a horizontal asymptote if as π‘₯ approaches positive or negative infinity, 𝑓 of π‘₯ tends to 𝑐. We can write these definitions for vertical and horizontal asymptotes in terms of limits.

For vertical asymptote, π‘₯ equals 𝑐 is vertical asymptote if either the limit as π‘₯ approaches 𝑐 from above of 𝑓 of π‘₯ is equal to positive or negative infinity or the limit as π‘₯ approaches 𝑐 from below of 𝑓 of π‘₯ is equal to positive or negative infinity. For horizontal asymptotes, we say that 𝑦 equals 𝑐 is a horizontal asymptote, if the limit as π‘₯ approaches infinity of 𝑓 of π‘₯ is equal to 𝑐 or the limit as π‘₯ approaches negative infinity of 𝑓 of π‘₯ is equal to 𝑐. Now that we have covered the definition of horizontal and vertical asymptote and the ways in which the different asymptote can occur, we’re ready to look at an example.

Determine the vertical and horizontal asymptotes of the function 𝑓 of π‘₯ is equal to negative one plus three over π‘₯ minus four over π‘₯ squared.

Let’s start by writing our function as one fraction. We find a common denominator of π‘₯ squared. And we can write 𝑓 of π‘₯ as negative π‘₯ squared plus three π‘₯ minus four over π‘₯ squared. In order to find the vertical asymptotes, we need to find the values of π‘Ž such that the limit as π‘₯ approaches π‘Ž from below of 𝑓 of π‘₯ is equal to plus or minus infinity or the limit as π‘₯ approaches π‘Ž from above of 𝑓 of π‘₯ is equal to plus or minus infinity. Since our 𝑓 of π‘₯ is irrational function, this will happen as the denominator of our function approaches zero.

Now, the denominator of our function is simply π‘₯ squared, so we can say that the vertical asymptotes will occur when π‘₯ squared is equal to zero. So we have found that there will be a vertical asymptote at π‘₯ equals zero. In order to find our horizontal asymptotes, we need to find the values of 𝑏 such that the limit as π‘₯ approaches infinity of 𝑓 of π‘₯ is equal to 𝑏. Or the limit as π‘₯ approaches negative infinity of 𝑓 of π‘₯ is also equal to 𝑏. So we take the limit of our function as π‘₯ approaches infinity. In order to find this limit, we need to multiply the top and bottom of our fraction by one over π‘₯ squared. And we obtained that the limit is π‘₯ approaches infinity of negative one plus three over π‘₯ minus four over π‘₯ squared all over one.

Next, we’ll be using the fact that the limit as π‘₯ approaches positive or negative infinity over one over π‘₯ is equal to zero. Therefore, three over π‘₯ and negative four over π‘₯ squared will both tend to zero. And so we find that our limit is left equal negative one. Therefore, we find we have a horizontal asymptote at 𝑦 is equal to negative one. We can quickly consider the limit as π‘₯ approaches negative infinity of our function. However, since the limit which we were using, that’s the limit as π‘₯ approaches positive or negative infinity of one over π‘₯ is equal to zero, works for both positive and negative infinity. We will see that the limit as π‘₯ approaches negative infinity gives us the same asymptote as the limit as π‘₯ approaches positive infinity.

So, therefore, we have found both the vertical and horizontal asymptotes of our function. It’s important to know that the function 𝑓 of π‘₯ may cross the asymptote at some point. We can see this if we sketch a graph of the function used in this example. Here, we can see a sketch of our function, including the horizontal and vertical asymptotes which we found, at π‘₯ equals zero and 𝑦 equals negative one. We can see that as π‘₯ tends to positive infinity, the function actually crosses the horizontal asymptote. However, it still shows asymptotic behavior since as π‘₯ gets larger and larger and larger, we can see that the line of the function is getting closer and closer to the line 𝑦 equals negative one. We can say that as π‘₯ goes to infinity, 𝑓 of π‘₯ gets arbitrarily close to 𝑦 equals negative one. Now, let’s look at another example.

What are the two asymptotes of the hyperbola 𝑦 is equal to five π‘₯ plus one over three π‘₯ minus four?

In order to find a vertical asymptote here, we need to find the values of 𝑏 such that any limit as π‘₯ approaches 𝑏 of 𝑦 is equal to positive or negative infinity. In order to find the vertical asymptotes, we simply need to find the values of π‘₯ such that the denominator of 𝑦 is equal to zero. What this mean is that three π‘₯ minus four is equal to zero. Rearranging this, we find that there is a vertical asymptote at π‘₯ is equal to four-thirds. In order to find the horizontal asymptotes of 𝑦, we need to consider the limit as π‘₯ goes to positive or negative infinity of 𝑦. In order to find the limit as π‘₯ approaches infinity of five π‘₯ plus one over three π‘₯ minus four, we first multiply the numerator and denominator of the fraction by one over π‘₯.

We are left with the limit as π‘₯ approaches infinity of five plus one over π‘₯ over three minus four over π‘₯. Then we can use the fact that the limit as π‘₯ approaches infinity of one over π‘₯ is equal to zero which tells us that one over π‘₯ and negative four over π‘₯ will both tend to zero as π‘₯ tends to infinity. And so, therefore, we find that our limit is equal to five-thirds. Let’s quickly note that if we consider the limit as π‘₯ approaches negative infinity of 𝑦, then we would see that this limit is also equal to five-thirds. Therefore, the solution to this question is that we have a vertical asymptote at π‘₯ equals four-thirds and a horizontal asymptote at 𝑦 equals five-thirds. Next, we’ll be considering a more general case in another example.

The graph of equation 𝑦 is equal to π‘Žπ‘₯ plus 𝑏 over 𝑐π‘₯ plus 𝑑 is a hyperbola only if 𝑐 is not on zero. In that case, what are the two asymptotes?

Let’s start by finding the vertical asymptote of this function. Vertical asymptotes occur at π‘₯ equals π‘˜ when either the limit as π‘₯ approaches π‘˜ from below of 𝑦 is equal to double negative infinity or the limit as π‘₯ approaches π‘˜ from above of 𝑦 is equal to positive or negative infinity. Since 𝑦 is a rational function, this will happen at π‘₯ values, where the denominator of 𝑦 is equal to zero. Therefore, we can find our vertical asymptotes by setting the denominator of 𝑦 equal to zero. Now, we solve 𝑐π‘₯ plus 𝑑 is equal to zero for π‘₯. We obtained that there’s a vertical asymptote as π‘₯ is equal to negative 𝑑 over 𝑐.

In order to find the horizontal asymptote of 𝑦, we need to consider the limit as π‘₯ approaches positive or negative infinity of 𝑦. In order to find this limit, we can multiply the numerator and denominator by one over π‘₯. Then we use the fact that the limit as π‘₯ approaches infinity of one over π‘₯ is equal to zero, in order to say that when we take this limit, 𝑏 over π‘₯ and 𝑑 over π‘₯ will both tend to zero. And this leaves us with π‘Ž over 𝑐. Now, we have found our horizontal asymptote. So we have found that the two asymptotes of our hyperbola are π‘₯ is equal to negative 𝑑 over 𝑐 and 𝑦 is equal to π‘Ž over 𝑐. We can use this result in order to help us quickly find asymptotes of hyperbolas of this form.

For example, if we want to find the asymptotes of the function 𝑦 is equal to nine π‘₯ minus 12 over five minus 12π‘₯. We can use the fact that our function of the form 𝑦 is equal to π‘Žπ‘₯ plus 𝑏 over 𝑐π‘₯ plus 𝑑 has asymptotes π‘₯ equals negative 𝑑 over 𝑐 and 𝑦 is equal to π‘Ž over 𝑐 in order to quickly find our asymptotes. Writing down the values of π‘Ž, 𝑏 𝑐, and 𝑑 might make things a bit easier. We find that our vertical asymptote is at π‘₯ is equal to five over 12. And our horizontal asymptote is at negative three over four. Now, let’s quickly note that it is possible for a function to have more than one vertical and one horizontal asymptote.

For example, let’s consider the function 𝑓 of π‘₯ is equal to one over π‘₯ squared minus four. We notice that the denominator of our function is a difference of two squares. And the function can, therefore, be written as one over π‘₯ plus two multiplied by π‘₯ minus two. Now in order to find our vertical asymptotes, we need to find the values of π‘₯ which will send our functions to infinity. In order to do this, we set the denominator of the function equal to zero. This will give us two asymptotes, one at π‘₯ is equal to negative two, which comes from π‘₯ plus two being equals zero, and the other at π‘₯ is equal to two, which will come from π‘₯ minus two being equal to zero.

If we take the limit of our function as π‘₯ goes to infinity, we can see that the denominator of this fraction will go to infinity as π‘₯ goes to infinity. Therefore, the value of this limit is simply zero which gives us a horizontal asymptote at 𝑦 is equal to zero, knowing the values of these three asymptotes will help us to sketch our graph. This is what the graph of our function will look like. As we can see, it has two vertical asymptotes and one horizontal. In the next example, we’ll be considering a case which we must be very careful of, which is when we have a factor within our rational function which can be cancelled. Let’s now look at the next example.

Find the asymptotes of the function 𝑓 of π‘₯ is equal to π‘₯ plus two over π‘₯ squared minus four.

Now, we’d normally start by finding the vertical asymptote by setting the denominator of our function equal to zero. However, we cannot do this immediately with this function. Let’s, instead, start by factorizing the denominator of 𝑓 of π‘₯. We can spot that we have a factor of π‘₯ plus two in both the numerator and denominator. And so we can cancel the factor in both the numerator and denominator here. However, we have to be careful since, in doing this, we will be slightly changing our function, since the original 𝑓 of π‘₯ is not defined as the value of π‘₯ is equal to negative two. However, our new function, which we can called 𝑔 of 𝑓 π‘₯, is. However, the asymptotes of 𝑓 and 𝑔 will still be the same.

Now in order to find the asymptotes of 𝑓, we simply need to find the asymptotes of 𝑔. In order to find the vertical asymptotes, we set the denominator of 𝑔 equal to zero. This gives us a vertical asymptote at π‘₯ equals two. In order to find the horizontal asymptotes, we need to find the limit of 𝑔 as π‘₯ goes to infinity. So that’s the limit as π‘₯ goes to infinity of one over π‘₯ minus two. Now, as π‘₯ goes to infinity, π‘₯ minus two will also go to infinity. And since π‘₯ minus two is in the denominator of the function here, that means that this limit will tend to zero. Therefore, we have a horizontal asymptote at 𝑦 equals zero. Let’s quickly sketch the graphs of 𝑔 of π‘₯ and 𝑓 of π‘₯, so that we can see how these two functions differ, despite sharing the same asymptotes.

As we can see from the sketches with the graph of 𝑔 of π‘₯ on the left and 𝑓 of π‘₯ on the right, the two functions look identical. The only difference is that, on the graph of 𝑓 of π‘₯, the point which has an π‘₯ value of negative two is undefined. Had we had tried to find the vertical asymptotes without cancelling the factor of π‘₯ plus two, then our denominator will be π‘₯ plus two multiplied by π‘₯ minus two. And in finding the vertical asymptote, we would have set this denominator equal to zero. And we would have said that there was two vertical asymptotes, one at π‘₯ equals negative two and one at π‘₯ equals two. However, as we can see from our graph, there’s no asymptote at π‘₯ equals negative two. All that we have is an undefined point. So, therefore, it’s very important to check that there’s no factors in our function which can be cancelled before we start finding asymptotes.

Now, we have seen a variety of examples of how we can find asymptotes and how useful asymptotes can be, especially when identifying or drawing graphs. We will now revisit some of the key points of this video. Key points. In order to find vertical asymptotes, we need to identify points which give a zero denominator. However, we must be careful to check if the rational function can simplify. In order to find horizontal asymptotes, we need to consider the limit of the function as π‘₯ tends to positive and negative Infinity. Asymptotes of a function can be useful to help us identify or sketch the graph of the function.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.