Horizontal Asymptote Rules for Rational Functions

For rational function R(x) = \frac{P(x)}{Q(x)} where P(x) has degree m and Q(x) has degree n: * If m < n: horizontal asymptote is y = 0 * If m > n: no horizontal asymptote exists * If m = n: horizontal asymptote is y = \frac{a}{b} where a and b are leading coefficients.

Example: f(x) = \frac{3x^2 + 1}{5x^3 - 2}: degree 2 < degree 3, so y = 0

g(x) = \frac{4x^3 + x}{2x^3 - 7}: equal degrees, so y = \frac{4}{2} = 2

The horizontal asymptote of a rational function depends on how the degrees of the numerator and denominator compare. When the denominator has higher degree, the function approaches zero as x approaches infinity.

Horizontal Asymptote Rules for Rational Functions is a Grade 11 math topic covered in enVision, Algebra 2 in Chapter 4: Rational Functions. Students at this level study the concept as part of their grade-level standards and are expected to explain, analyze, and apply what they have learned.

f(x) = \frac{3x^2 + 1}{5x^3 - 2}: degree 2 < degree 3, so y = 0; g(x) = \frac{4x^3 + x}{2x^3 - 7}: equal degrees, so y = \frac{4}{2} = 2; h(x) = \frac{x^3 + 5}{x^2 - 1}: degree 3 > degree 2, so no horizontal asymptote