SECTION 4.5 Properties of Rational Functions 243 Finding a Horizontal or an Oblique Asymptote Find the horizontal or oblique asymptote, if one exists, of the graph of ( ) = − + − R x x x x 8 2 4 1 2 2 Solution EXAMPLE 7 Since the degree of the numerator, 2, equals the degree of the denominator, 2, the rational function R has a horizontal asymptote equal to the ratio of the leading coefficients. = = = y a b 8 4 2 n m To see why the horizontal asymptote equals the ratio of the leading coefficients, investigate the behavior of R as →−∞ x or as →∞ x . When x is very large, the numerator of R can be approximated by the power function = y x8 ,2 and the denominator can be approximated by the power function = y x4 .2 This means that as →−∞ x or as →∞ x , ( ) = − + − ≈ = = R x x x x x x 8 2 4 1 8 4 8 4 2 2 2 2 2 The graph of the rational function R has a horizontal asymptote = y 2. The graph of R resembles the graph of = y 2 as →±∞ x . Check: Verify the results by creating a TABLE with ( ) = Y R x 1 and = Y 2. 2 As a result, ( ) = − − + = + + − − − + H x x x x x x x x x x 3 1 3 3 2 3 3 1 4 2 3 2 2 3 2 As →−∞ x or as →∞ x , − − − + ≈ = → x x x x x x x 2 3 3 1 2 2 0 2 3 2 2 3 As →−∞ x or as →∞ x , we have ( ) → + H x x3 3. See Table 17 with ( ) H x and = + y x3 3. 1 As →∞ x or →−∞ x , the difference in values between ( ) H x and y1 becomes indistinguishable.The graph of the rational function H has an oblique asymptote = + y x3 3. Put another way, as →±∞ x , the graph of H resembles the graph of = + y x3 3. Figure 41 shows the graph of ( ) = − − + H x x x x x 3 1 4 2 3 2 and the graph of = + y x3 3 1 using Desmos. Table 17 Finding a Horizontal or an Oblique Asymptote Find the horizontal or oblique asymptote, if one exists, of the graph of ( ) = − + − G x x x x 2 2 1 5 3 3 EXAMPLE 8 Figure 41 ( ) = − − + H x x x x x 3 1 4 2 3 2 (continued)
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