1. Table of Contents Rational Functions: Horizontal Asymptotes Horizontal Asymptotes: A horizontal asymptote of a rational function is a horizontal line (equation: y = number) such that as values of the independent variable, x, decrease without bound or increase without bound, the function values (y-values) approach (get closer and closer to) the number (from either above or below). The next two slides illustrate the definition.

2. Table of Contents Rational Functions: Horizontal Asymptotes Slide 2 The rational function shown graphed has horizontal asymptote: y = 2 (dashed line), since as x-values decrease without bound, y-values approach 2 from "below". -30 -20 -10 0 10 20 30 -2246 x decreasing without bound y approaches 2 from "below" x- 1- 10- 100 - 1000 y0.51.54 1.94 1.994

3. Table of Contents Rational Functions: Horizontal Asymptotes Slide 3 Also note as x-values increase without bound, y-values approach 2 from "above". -30 -20 -10 0 10 20 30 -2246 x increasing without bound y approaches 2 from "above" x 6 10 100 1000 y 42.86 2.06 2.006

4. Table of Contents Rational Functions: Horizontal Asymptotes Slide 4 Example 1: Algebraically find the horizontal asymptote of, First, compare the degree of the polynomial in the numerator with the degree of the polynomial in the denominator. There are 3 possibilities to consider. (1)If degree of numerator < degree of denominator, the horizontal asymptote is y = 0 (the x-axis). (2)If degree of numerator > degree of denominator, there is no horizontal asymptote.

5. Table of Contents Rational Functions: Horizontal Asymptotes Slide 5 (3)If degree of numerator = degree of denominator, the horizontal asymptote is In the function, the degrees of the polynomials in both the numerator and denominator are 1, so the horizontal asymptote is found using (3) above. This function’s horizontal asymptote is y = 2. (This was the function shown graphed in the preceding slides!)

6. Table of Contents Rational Functions: Horizontal Asymptotes Slide 6 This horizontal asymptote is y = 0. Try: Algebraically find the horizontal asymptote of, This horizontal asymptote is y = 1/4. Try: Algebraically find the horizontal asymptote of,

7. Table of Contents Rational Functions: Horizontal Asymptotes