SECTION 4.5 Properties of Rational Functions 241 For a rational function in lowest terms, the multiplicities of the real zeros in the denominator can be used in a similar fashion to determine the behavior of the graph around each vertical asymptote. Consider the following Exploration. Exploration Graph each of the following rational functions using a graphing utility and describe the behavior of the graph near the vertical asymptote = x 1. ( ) ( ) ( ) ( ) ( ) ( ) ( ) = − = − =− − =− − R x x R x x R x x R x x 1 1 1 1 1 1 1 1 1 2 2 3 3 4 4 Result For R ,1 as x approaches 1 (odd multiplicity) from the left side of the vertical asymptote, the values of ( ) R x 1 approach negative infinity. That is, ( ) =−∞ → − R x lim . x 1 1 As x approaches 1 from the right side of the vertical asymptote, the values of ( ) R x 1 approach infinity. That is, ( ) =∞ → + R x lim . x 1 1 See Figure 40(a). Below, we summarize the behavior near = x 1 for the remaining rational functions: • For R :2 ( ) =∞ → − R x lim x 1 2 and ( ) =∞ → + R x lim x 1 2 [See Figure 40(b)] • For R :3 ( ) =∞ → − R x lim x 1 3 and ( ) =−∞ → + R x lim x 1 3 [See Figure 40(c)] • For R :4 ( ) =−∞ → − R x lim x 1 4 and ( ) =−∞ → + R x lim x 1 4 [See Figure 40(d)] 6 26 23 5 (b) R2(x); even multiplicity 6 26 23 5 (c) R3(x); odd multiplicity Figure 40 6 26 23 5 (a) R1(x); odd multiplicity 6 26 23 5 (d) R4(x); even multiplicity The results of the Exploration lead to the following conclusion. Multiplicity and Vertical Asymptotes • If the multiplicity of the real zero that gives rise to a vertical asymptote is odd, the graph approaches ∞ on one side of the vertical asymptote and approaches −∞ on the other side. • If the multiplicity of the real zero that gives rise to the vertical asymptote is even, the graph approaches either ∞ or −∞ on both sides of the vertical asymptote. Now Work PROBLEMS 45, 47, AND 49 (FIND THE VERTICAL ASYMPTOTES, IF ANY) 3 Find the Horizontal or Oblique Asymptote of a Rational Function To find horizontal or oblique asymptotes, we need to know how the value of the function behaves as →−∞ x or as →∞ x . That is, we need to determine the end behavior of the function.This can be done by examining the degrees of the numerator and denominator, and the respective power functions that each resembles. For example, consider the rational function ( ) = − − + R x x x x 3 2 5 7 1 2 The degree of the numerator, 1, is less than the degree of the denominator, 2. When x is very large, the numerator of R can be approximated by the power function = y x3 , and the denominator can be approximated by the power function = y x5 .2 This means R x x x x x x x 3 2 5 7 1 3 5 3 5 0 2 2 ( ) = − − + ≈ = → ↑ ↑ For x very large As x →−∞ or x →∞ which shows that the line = y 0 is a horizontal asymptote.We verify this in Table 16. This result is true for all rational functions that are proper (that is, the degree of the numerator is less than the degree of the denominator). If a rational function is improper (that is, if the degree of the numerator is greater than or equal to the degree Table 16
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