Asymptote: Definition & Meaning

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  • 0:00 What is an Asymptote?
  • 2:11 Horizontal Asymptote
  • 2:35 Vertical Asymptotes
  • 2:56 Slant Asmptotes
  • 3:18 Lesson Summary
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Lesson Transcript
Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

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.

Your journey to your car
1/2 1/4 1/8 pattern

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.

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