Family Overview

In Plain Language

A vertical asymptote is a vertical line $x=a$ that the graph of a rational function approaches while outputs blow up to $\infty$ or $-\infty$.

For $r(x)=\dfrac{p(x)}{q(x)}$:

• If $a$ is a real zero of the denominator but not of the numerator, then $x=a$ is a vertical asymptote.
• If $a$ is a zero of both, compare multiplicities: a VA still occurs when the multiplicity in the denominator is greater than in the numerator.
• Near $x=a$, one-sided limits describe the blow-up: e.g. $\lim_{x\to a^+}r(x)=\infty$ or $-\infty$ (and similarly from the left).

(If multiplicities make the factor cancel enough, you may get a hole instead—Topic 1.10.)

Why It Matters

Vertical asymptotes explain the “breaks” in rational graphs and connect domain restrictions to unbounded local behavior. AP items ask students to locate VAs and describe nearby limits—not confuse them with zeros (1.8) or horizontal asymptotes (1.7).

How Families Can Help

• Ask: “Does this denominator zero also zero the numerator?”
• Practice multiplicity compare: den mult > num mult ⇒ VA.
• Have them say one-sided limits out loud near a VA.

What Not to Do

• Do not call every denominator zero a VA without checking the numerator.
• Do not mix vertical asymptotes with horizontal end behavior.
• Do not treat a zero of $r$ as a VA.
• Do not skip one-sided limit language when describing “blows up.”