Asymptotes - Online Tutor, Practice Problems & Exam Prep

Asymptotes are imaginary lines that graphs approach but never touch or cross. Horizontal asymptotes, like \(y = 2\) or \(y = 0\), indicate values the graph approaches as it extends horizontally. Vertical asymptotes, such as \(x = -2\) and \(x = 2\), occur where the graph becomes steep and approaches a value without reaching it. Understanding these behaviors is crucial for analyzing polynomial functions and their limits, guiding the overall behavior of the graph.

Asymptotes

Everyone. Welcome back. So, one thing that you may have learned in math classes up until now is how to draw the graphs of equations and functions and look at their behaviors. One very common behavior that you'll see in a graph is where a graph gets really, really close to a value, but it never actually quite touches it. Look at this purple graph, and as it gets all the way out to the right, you'll see that it gets close to a value of 2, but it never actually quite gets there. This is a perfect example of what we call an asymptote. I'm going to go ahead and review asymptotes in this video because you'll need to be able to identify them and also draw them and write their equations. We'll do some examples together. Let's get started here. An asymptote is really just like kind of like an imaginary line that a graph gets really close to, but it never actually touches or crosses. So, this purple graph over here gets really close as we go to the right to a value of 2, but it never actually touches it or crosses that barrier. Notice how if we keep extending this purple line, it'll get really, really, really close to 0, like 0.0001, but never actually gets there. So, this is an example of a horizontal asymptote. We draw this with a sort of dashed line like this just to kind of separate it from the graph. And what you'll notice here is that on the opposite side of the graph, the same thing actually happens. If you look at the bottom now, this graph is going to approach this value from the left side. So as it goes off to the left, it gets really flat, never quite gets to 2, but it gets really, really, really close. So we draw this as a horizontal asymptote. Now what's the equation for it? Well, these things are usually just going to be vertical or horizontal lines, so we just write them as horizontal or vertical line equations. Horizontal lines are written as y = 2. So that's the horizontal asymptote.

Now, if you actually look at this graph here, there's actually another example of this behavior happening. Look at the tops and bottom of this graph. Notice how as we come in from the left, this purple graph gets really, really, really steep. It gets really vertical. It never actually quite gets to a value of x = 1, but it gets really close. And then, once we cross that barrier over here on the other side, we'll see that now the graph is approaching it's coming in from the top all the way down like this. So what happens usually at these asymptotes is that it kind of sets up like a barrier in which the graph can't cross it, but it'll kind of go off to infinity in each direction, but you'll see the graph coming in from opposing sides like this. Very common behavior for asymptotes. And in this particular case, the vertical asymptote is x = 1. That's really all there is to it. We just draw them as dashed lines and then write their equations.

Let's take a look at the next example over here. We're going to have this other purple graph, and we're going to identify and sketch all the asymptotes of the function below. Notice how it doesn't actually matter what this function is. All we really need to do is just figure out and identify where these asymptotes are happening. So let's take a look for any horizontal asymptotes here. This purple graph, you'll see, goes off to the right and the left, kind of like our graph above did. And, usually, that's a good place to look for asymptotes. Notice how on the right side, this thing gets really flat, and it approaches a value of 0, but never actually quite gets there. On the left side, the same thing happens. It approaches a value of 0, but never quite gets there. So, there is a horizontal asymptote here at y = 0. It's perfectly fine for asymptotes to be on the x or y-axis. That actually very commonly happens. There aren't any other horizontal asymptotes here because notice how the graph doesn't get close to a horizontal or a value as it goes off to either end, and we're crossing all of the other values over here.

So let's take a look for any vertical asymptotes now. What you see here is that the graphs get really steep at certain values, and you'll notice that as the graph comes in from the left, it goes really vertical and never actually quite gets to a value of x = -2, and the graph would just sort of go off to infinity there. Then what happens is on the other side, let's take a look at this sort of middle upside-down U over here. As we go off to the left over here, you'll see that this graph also approaches a value of -2. So this is going to be x = -2. That's the vertical asymptote. There's actually one more over here because this graph ends up being really symmetrical. You'll see that the same exact behavior happens over here on the right side. This graph gets really close to a value of 2, never quite gets there. And on the other side of this U, it gets close to a value of 2, but never quite gets there. So this is a vertical asymptote at x = 2. That's it. These are the 3 asymptotes that guide the behavior of this graph. That's really all you need to know. Thanks for watching. Let me know if you have any questions.

Here’s what students ask on this topic:

What is an asymptote in mathematics?

An asymptote is an imaginary line that a graph approaches but never actually touches or crosses. Asymptotes can be horizontal, vertical, or oblique. For example, a horizontal asymptote might be represented by the equation $y=2$, indicating that as the graph extends horizontally, it gets closer to $2$ but never reaches it. Vertical asymptotes, such as $x=1$, occur where the graph becomes infinitely steep and approaches a specific value without touching it. Understanding asymptotes is crucial for analyzing the behavior of functions and their limits.

How do you find horizontal asymptotes of a function?

To find horizontal asymptotes of a function, you need to analyze the behavior of the function as $x$ approaches positive or negative infinity. For rational functions, compare the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y=0$. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote.

What is the difference between a vertical and a horizontal asymptote?

A vertical asymptote occurs where the graph of a function becomes infinitely steep and approaches a specific value of $x$ without touching it, such as $x=1$. A horizontal asymptote, on the other hand, indicates a value that the graph approaches as $x$ extends towards positive or negative infinity, like $y=2$. Vertical asymptotes are found by setting the denominator of a rational function to zero, while horizontal asymptotes are determined by analyzing the end behavior of the function.

How do you determine the equation of a vertical asymptote?

To determine the equation of a vertical asymptote, you need to find the values of $x$ that make the denominator of a rational function equal to zero, provided these values do not cancel out with the numerator. For example, for the function $\frac{f}{g}$, if $g\left(x\right)=0$ at $x=1$, then $x=1$ is a vertical asymptote. The equation of the vertical asymptote is $x=1$.

Can a function have more than one horizontal asymptote?

No, a function cannot have more than one horizontal asymptote. A horizontal asymptote represents the value that the function approaches as $x$ goes to positive or negative infinity. Since a function can only approach one value in each direction, it can have at most one horizontal asymptote. However, a function can have different horizontal asymptotes as $x$ approaches positive infinity and negative infinity, but this is rare and typically occurs in piecewise functions.