AP Calculus AB and BC

Vertical vs Horizontal Asymptote

A vertical asymptote occurs where a one-sided limit is infinite, and a graph can never cross one. A horizontal asymptote comes from a finite limit as the input grows without bound, describes long-run behaviour only, and can be crossed any number of times.

Vertical asymptote

Use when: You are looking at behaviour near a specific input where the function blows up.

Horizontal asymptote

Use when: You are looking at end behaviour as the input grows without bound.

Side by side

	Vertical	Horizontal
Comes from	$\lim_{x \to a} f(x) = \pm\infty$	$\lim_{x \to \pm\infty} f(x) = L$
Equation	$x = a$	$y = L$
Can be crossed	No	Yes
How many possible	Any number	At most two

A graph cannot cross a vertical asymptote because there is no function value there at all. A horizontal asymptote is only a statement about the tail, so crossing it near the origin is perfectly ordinary.

For rational functions, find vertical asymptotes at the denominator zeros that survive cancelling, and horizontal ones by comparing degrees: smaller on top gives $y = 0$ , equal degrees give the ratio of leading coefficients.

The direction trap

As $x \to -\infty$ , $\sqrt{x^2}$ equals $-x$ , not $x$ . Forgetting that flips the sign and is why some functions have two different horizontal asymptotes.

Frequently asked questions

Why at most two horizontal asymptotes?

Because there are only two directions to travel, so there are only two limits at infinity to take.

In the CED: Unit 1: Limits and Continuity