Infinity is a hard concept to understand and the word asymptote is pretty intimidating.
But this fun lesson will make both seem like a walk in the park as it defines both and shows their relationship using a graph.
Genie in a Bottle
If a genie popped out of a bottle and granted you just one wish, what would that be for you? Perhaps you would wish for a puppy. Maybe you’d wish you had a Lamborghini. But the best wish of all would be to have an infinite number of wishes! Why? You’re about to find out as you learn about asymptotes and infinity.
What is Infinity?
The genie in the bottle would probably not be really happy at not qualifying the granting of your wish. He probably should’ve said you could have one wish, so long as it’s not a wish for an infinite number of wishes. Why? That’s because infinity refers to something that has no end, is unlimited, and is unbounded.
In other words, there is no limit to infinity. Infinity is so cool it even has a funky symbol for it.Anyway, with that definition you can see why the genie is a bit peeved.
Since he has to grant you your wish, he’ll have to grant you an unending number, an infinite number of wishes! Now you can have that puppy, and that Lamborghini, and anything else your heart desires.
What is an Asymptote?
One of the things that can lead off into infinity is a curve that is drawn on a graph. You can see here that the curve is getting closer and closer to the green line, labeled as the horizontal asymptote. An asymptote is a line that serves as the limit of a curve. The word comes from the Greek asymptotes, which means not falling together.In other words, as the curve approaches this line it gets closer and closer to the asymptote but never quite reaches it.
This means that as the curve and line head off into infinity, the distance between the two approaches ever closer to – but never actually reaches – zero.There are three main kinds of asymptote. The first one you just learned about is the horizontal asymptote. The other two aren’t any harder to understand.
One is the vertical asymptote. This is a vertical line. Here, again, you can see that as the curve is drawn, it approaches but never quite touches the vertical asymptote. The other kind of asymptote is the oblique one. This is simply a slanted line that is approached by the curve.
Don’t you wish this lesson was over? Well thanks to your infinite number of wishes, the Genie has made your wish come true. So, let’s review everything we just learned. Infinity refers to something that has no end, is unlimited, and is unbounded. A curve and a line on a graph can head off into infinity. As a curve heads off to infinity, it may come close to but never reach an asymptote, which is a line that serves as the limit of a curve.
Asymptotes can be horizontal, vertical, or oblique.The whole point is that as both the asymptote and curve head off into infinity, the distance between the two gets smaller and smaller, meaning the distance between them gets closer and closer to zero. However, since the asymptote serves as a sort of boundary for the curve, the curve never truly reaches this boundary, but it does get infinitely close.