242 CHAPTER 4 Polynomial and Rational Functions of the denominator), there could be a horizontal asymptote, an oblique asymptote, or neither.A rational function will not have both a horizontal asymptote and an oblique asymptote. However, a rational function may have neither a horizontal nor an oblique asymptote. The following details how to find horizontal or oblique asymptotes. Finding a Horizontal or an Oblique Asymptote of a Rational Function Consider the rational function ( ) ( ) ( ) = = + + + + + + + + − − − − R x p x q x a x a x a x a b x b x b x b n n n n m m m m 1 1 1 0 1 1 1 0 in which the degree of the numerator is n and the degree of the denominator is m. • If < n m (the degree of the numerator is less than the degree of the denominator), the line = y 0 is a horizontal asymptote. • If = n m (the degree of the numerator equals the degree of the denominator), the line = y a b n m is a horizontal asymptote. (That is, the horizontal asymptote equals the ratio of the leading coefficients.) • If = + n m 1 (the degree of the numerator is one more than the degree of the denominator), the line = + y ax b is an oblique asymptote, which is the quotient found using polynomial division. • If ≥ + n m 2 (the degree of the numerator is two or more greater than the degree of the denominator), there are no horizontal or oblique asymptotes, and the end behavior of the graph resembles the power function = − y a b x . n m n m Note: Rational functions that simplify to a linear function do not have asymptotes. Finding a Horizontal Asymptote Find the horizontal asymptote, if one exists, of the graph of ( ) = − + + − R x x x x x x 4 5 2 7 2 3 3 5 4 Solution EXAMPLE 5 Since the degree of the numerator, 3, is less than the degree of the denominator, 5, the rational function R is proper. The line = y 0 is a horizontal asymptote of the graph of R. Finding a Horizontal or an Oblique Asymptote Find the horizontal or oblique asymptote, if one exists, of the graph of ( ) = − − + H x x x x x 3 1 4 2 3 2 Solution EXAMPLE 6 Since the degree of the numerator, 4, is exactly one greater than the degree of the denominator, 3, the rational function H has an oblique asymptote. Find the asymptote by using polynomial division. ) + − + − − + − − − + − − x x x x x x x x x x x x x x x 3 3 1 3 3 3 3 3 3 3 3 3 2 3 3 3 2 4 2 4 3 3 2 3 2 2
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