Rational Functions and Asymptotes

Vertical Asymptotes

Where the denominator = 0 (and numerator ≠ 0). $f(x) = \frac{1}{x-2}$: VA at $x = 2$.

Horizontal Asymptotes

Compare degrees:

• Degree(num) < Degree(den): $y = 0$.
• Degree(num) = Degree(den): $y = \frac{\text{leading coefficients}}{}$.
• Degree(num) > Degree(den): no HA (may have oblique).

Holes

Common factor in numerator and denominator creates a hole, not an asymptote.

$f(x) = \frac{2x+1}{x-3}$. VA: $x = 3$. HA: $y = 2$ (same degree, ratio of leading coefficients).

X-intercept: $2x + 1 = 0$ → $x = -\frac{1}{2}$. Y-intercept: $f(0) = -\frac{1}{3}$.