Horizontal and Vertical Asymptotes

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  • 0:49 Asymptotes
  • 1:51 Horizontal Asymptotes
  • 4:04 Vertical Asymptotes
  • 7:01 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

No matter how hard you try to get to them, asymptotes remain out of reach. Learn about these invisible lines on graphs that show you places your equations just can't go.

Running to the Finish Line

Regardless of how large x gets, y will never reach 0
Asymptote Race Example

There's this old paradox that says that to get to any finish line, you first have to get halfway there, no matter where you start. If you're running a race, you first have to get halfway to the finish before you can actually get to the finish. But, when you're halfway to the finish, you have to get halfway of what's left to finish, so that you're three-quarters of the way to the finish. No matter how close you get to the finish line, you can only get halfway to the finish from where you're at. Eventually, this gets very frustrating. You have to wonder, is the finish line is just out of reach?


Let's talk a little bit about asymptotes. Like our finish line that's out of reach, these are places that you can't quite get to. For example, if we look at distance to our goal, or distance to the finish line, as a function of time, every second we get halfway to the goal. We get closer and closer to this horizontal x-axis. But we never quite get to the finish line.

Mathematically, we could write this as, say, y=1/x. y is the distance to the finish line and x is the amount of time that we've spent trying to get to the finish line. No matter how big x gets, y will never actually get to zero. No matter how long you keep going, you can't quite get to the finish line. You're always going to be just next to it. But if you're just next to it, your next step can only take you halfway to the finish line, which still isn't there.

Let's take a look at an example: y = x/(x^2 + 1) + 1. Let's graph this out.

Graph for y = x/(x^2 + 1) + 1
Horizontal Asymptotes Example Graph

Horizontal Asymptotes

Let's define one of these horizonal asymptotes. If y approaches some number, like y goes to N as x goes to +/- infinity, then the line y=N is a horizontal asymptote. In the case of y=1/x, y is approaching zero as x gets really big, as x goes to + infinity. This is like saying that as time goes to infinity, the distance to the goal is going to zero, but it's not actually going to hit zero. You're not actually going to get there.

This graph looks kind of like a heartbeat. As x gets really, really large, we move to the far right side of this graph. And indeed, we can keep going far beyond this graph, as far as you can imagine. As x gets really large as we go as far as you can imagine to the right, y is going to approach the value of 1. On the other side, as x gets really, really small, as we move far to the left in x, y is also going to approach 1. Although it's going to approach from below, whereas for the positive values of x, it's going to approach 1 from above. Let's look at this right-hand side in more detail. When x is equal to 1, y is 1.5. As I increase x to 5, y slowly approaches 1; it's 1.19. y is 1.10 when x is 10. When x is 100, y is 1.01. And, at x=1,000, y is 1.001. As you can see, as x gets larger and larger, y is approaching 1. But it's never actually going to reach 1. You can't solve this equation for y=1.

With horizontal asymptotes, the key is what happens as your independent variable gets very large, either large in the positive direction or very large but negative.

As x gets closer to equaling 1 on the left side of the graph, y gets larger and larger
Vertical Asymptote Graph

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