Sometimes, a graph of a curve will get closer and closer to a line without ever reaching it. This line is called an asymptote, and it can be horizontal, vertical, or slanted.

## What Is an Asymptote?

An **asymptote** is a value that you get closer and closer to, but never quite reach. In mathematics, an asymptote is a horizontal, vertical, or slanted line that a graph approaches but never touches.

Imagine that you are walking back to your car that is parked at the farthest corner of the mall parking lot. After a long day of shopping, you are exhausted. After about a minute of walking you are halfway to your car, but you notice you are gradually slowing down. Another minute later, you have only walked another 1/4 of the way to your car. Your legs continue to feel heavier and heavier. After a third minute, you are only another 1/8 of the way closer. The pattern continues. As your shopping-exhausted body drags even more and more, it seems like you will never get there. Each additional minute only brings you halfway to the car from where you were previously.

Well, here is the surprise: If this pattern continues, then, in theory, you won’t ever get there. Yes, you get closer and closer to your car by the minute, but the distances you travel get smaller and smaller. With time you get really, really close. You get so close you can touch your car. You get so close that your steps are smaller than the head of a pin, but you are always only covering half the remaining distance to the car.

This situation, often called **Zeno’s paradox**, is a bit silly in reality – at some point, no matter how tired you are, you step more than half the remaining distance to your car. The amount you have left to travel is less than one step, so you just take it and arrive; however, mathematicians often like to think in the world of theory.

Suppose, for instance, that there is a force field surrounding your car. As you get closer to the car, it becomes harder and harder to move closer. It now requires an enormous amount of effort to take each step, so they become smaller. You literally cannot reach your car because, once you are really close to the car, the force it would take to get all the way there is too strong for you to overcome. This is how the mathematician thinks about asymptotes: There is a place that a graph can get closer and closer to but never quite reach.

## Horizontal Asymptotes

A **horizontal asymptote** occurs when a graph can’t reach some horizontal line (*y* can’t equal some value). That line might be the *x*-axis.

But, there can also be a horizontal asymptote somewhere else. In the figure below, there is a horizontal asymptote at *y* = 3. In other words, *y* can never quite equal 3, but it gets closer as *x* gets bigger.

## Vertical Asymptotes

A **vertical asymptote** occurs when *x* can’t equal some value. For example, if you graph *y* = 1/*x*, you will see that *x* can’t ever equal zero. So, there is a vertical asymptote at *x* = 0. There is also a horizontal asymptote at *y* = 0. A graph can have both kinds of asymptotes.

## Slant Asymptotes

Sometimes, there is a slanted, or diagonal, line that a curve approaches but never quite reaches. One time this happens is when there is an *x*-squared in the numerator of a fraction and an *x* in the denominator. For this example, the figure below shows a slant asymptote of the line *y* = *x* – 3. This graph also has a vertical asymptote.

## Lesson Summary

An **asymptote** is a line that a graph approaches but never touches. That line can be vertical, horizontal, or even slanted, and graphs can have multiple types of asymptotes.