AP Precalculus / Unit 1: Polynomial and Rational Functions
1.9 Rational Functions and Vertical Asymptotes
Notes
A vertical asymptote at $x=a$ happens when $a$ is a zero of the denominator but not the numerator (or has higher multiplicity in the denominator).
- Near the asymptote, $|r(x)|$ grows without bound.
Practice
Vertical asymptotes of $\displaystyle \frac{x}{x^2-4}$:
The bottom is 0 at $\pm2$; the top isn't.
$\displaystyle \frac{x-1}{(x-1)^2}$ has at $x=1$:
The denominator's factor has higher multiplicity.
As $x\to3^+$, $\displaystyle \frac{1}{x-3}\to$
Small positive denominator.
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