Infinity and Asymptotes: Exploring Limits and Vertical Asymptotes

Limits Involving Infinity

In this section, we’ll look at limits to infinity, positive infinity, and horizontal and vertical asymptotes. When we talk about horizontal asymptotes and limits to infinity, some functions might not have a horizontal asymptote. lim_(x-> infinity) f(x) [Graph] lim_(x-> -infinity) f(x) [Graph] [Graph] __________________________________________________ [Graph][Graph] lim_(x->a) f(x) [Graph] [Graph] A function could be represented by this equation. And as x approaches infinity, the value of y approaches 0.

But instead of the asymptote being at 0, we could have an asymptote along any other y value. We’ll call c this new value. So in this case, the limit as x approaches infinity (it could be negative infinity as well) of this function we call f of x is going to be c. Now, in addition to horizontal asymptotes (which have limits to positive or negative affinity), we want to discuss vertical asymptotes (which have an affinity as a limit).

If we approach a finite value for x, which in this case we’ll say a, there are four possibilities that may happen. Our function from the left hand side and the right hand side could be increasing without bound, in which case we would say the limit is positive infinity. If the left and right sides of the asymptote go down without bound, the function is said to go to negative infinity. We could have some sort of disagreement, in which case one side may increase its outbound flow, while the other decreases its outbound flow.

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The limit does not exist because the left and right hand side do not agree. These are the four possibilities for vertical asymptotes.