Topic 1.7 - Rational Functions and End Behavior

Case I: n = d

• If the leading terms have the same degree, the horizontal asymptote is: $y = \frac{a}{b}$

Case II: n < d

• If the denominator dominates, the horizontal asymptote is: $y = 0$

Case III: n > d

• End behavior: $f(x) \sim \frac{a}{b} x^{n-d}$ (dominant term behaves like this polynomial)

• Note: If the degree difference is 1 (n = d + 1), there is a slant (oblique) asymptote.

• Slant asymptote direction: parallel to the line $y = \frac{a}{b} x$

Rational Function basics

• Rational Function: $y = \frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are polynomials and $g(x) \neq 0$
• Examples:
  • $y = \frac{2}{x+3}$
  • $y = \frac{x^{3} - 3x + 1}{3x + 4}$
  • $y = \frac{2x^{3} + 4x - 6}{x^{2} - 7x + 11}$

Example 1: Determine horizontal/slant end behavior

• a) $f(x) = \frac{3x^{2} + 4x - 7}{5x^{2} - 3}$
  • Degrees equal ⇒ horizontal asymptote: $y = \frac{3}{5}$
• b) $y = \frac{2x - 5}{x^{2} + 3x + 2}$
  • Denominator degree > numerator ⇒ horizontal asymptote: $y = 0$
• c) $g(x) = \frac{2x^{2} - 4}{5x + 9}$
  • Numerator degree 2, denominator degree 1 ⇒ slant asymptote
  • Slope of slant: $\frac{a}{b} = \frac{2}{5}$
  • Exact line (from long division): $y = \frac{2}{5}x - \frac{18}{25}$
• d) $y = \frac{4x + 5}{8x - 1}$
  • Degrees equal ⇒ horizontal asymptote: $y = \frac{4}{8} = \frac{1}{2}$
• e) $k(x) = \frac{3}{x^{2} + 3x - 7}$
  • Numerator degree 0, denominator degree 2 ⇒ horizontal asymptote: $y = 0$
• f) $p(x) = \frac{-4}{2x + 1}$
  • Horizontal asymptote: $y = 0$

Example 2: End behavior limits

• a) $f(x) = \frac{2x^{4} + 4x - 1}{6x^{4} - x^{3} + 4}$
  • Leading terms ratio: $\frac{2}{6} = \frac{1}{3}$
  • Thus: $\lim<em>{x \to \infty} f(x) = \lim</em>{x \to -\infty} f(x) = \frac{1}{3}$
• b) $g(x) = \frac{5x^{4} - 8x + 9}{2x^{4} + x - 1}$
  • Leading terms ratio: $\frac{5}{2}$
  • Thus: $\lim<em>{x \to \infty} g(x) = \lim</em>{x \to -\infty} g(x) = \frac{5}{2}$
• c) $h(x) = \frac{-3x^{2} - x^{2} + x}{x^{2} + 4x + 4}$
  • Leading terms ratio: $\frac{-3}{1} = -3$
  • Thus: $\lim_{x \to \pm \infty} h(x) = -3$
• d) (content not provided in the transcript)
• e) (content not provided in the transcript)

Example 3: Slant asymptote parallel to y = (1/2) x

• A slant asymptote exists when the numerator degree is exactly one more than the denominator degree, and its slope is the leading ratio $\frac{a}{b}$
• I. $f(x) = \frac{x^{2} + 3}{2x^{2} + x + 6}$ → Degrees equal ⇒ no slant
• II. $g(x) = \frac{x^{3} + 4x + 1}{2x^{5} + x^{2} + 2}$ → Denominator degree 5 > numerator ⇒ no slant
• III. $h(x) = \frac{x^{2} + 3x + 5}{2x + 4}$ → Degrees 2 and 1; n = d + 1 ⇒ slant; slope $\frac{a}{b} = \frac{1}{2}$
• IV. $k(x) = \frac{x^{2} + x + 5}{2x^{2} + x - 1}$ → Degrees equal ⇒ no slant
• Answer: III only (option C).

Slant asymptotes recap

• If the numerator degree is exactly 1 greater than the denominator degree, the slant asymptote is parallel to the line $y = \frac{a}{b} x$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.