Rational Functions and Asymptotes

Rational Functions

$f(x) = \frac{p(x)}{q(x)}$

Vertical Asymptotes and Holes

Factor both numerator and denominator:

• Hole: Common factor cancels → point discontinuity
• Vertical Asymptote (VA): Factor remains in denominator

$f(x) = \frac{(x-2)(x+1)}{(x-2)(x-3)}$

Hole at $x = 2$; VA at $x = 3$

Behavior Near VAs

$\lim_{x \to 3^+} f(x) = +\infty \quad \text{or} \quad -\infty$ $\lim_{x \to 3^-} f(x) = +\infty \quad \text{or} \quad -\infty$

Horizontal Asymptotes (HA)

$f(x) = \frac{a_nx^n + \cdots}{b_mx^m + \cdots}$

| Condition | HA | |-----------|-----| | $n < m$ | $y = 0$ | | $n = m$ | $y = \frac{a_n}{b_m}$ | | $n > m$ | No HA |

$\lim_{x \to \pm\infty} f(x) = \text{HA value}$

Slant (Oblique) Asymptotes

When $n = m + 1$, perform polynomial division. The quotient is the slant asymptote.

Zeros of Rational Functions

Set the numerator equal to zero (after canceling common factors): $\frac{p(x)}{q(x)} = 0 \iff p(x) = 0$

Solving Rational Inequalities

1. Find zeros and undefined values
2. Create a sign chart
3. Test intervals

Partial Fractions

$\frac{2x + 5}{(x+1)(x+3)} = \frac{A}{x+1} + \frac{B}{x+3}$

Solve for $A$ and $B$ by substituting convenient values.

AP Precalculus Tip: Always use limit notation when describing asymptotic behavior. A function "approaches" an asymptote; it doesn't "equal" it.