SECTION 4.5 Properties of Rational Functions 239 The results of the Exploration reveal an important property of rational functions. The vertical line = x 2 and the horizontal line = y 1 in Figure 37(c) are called asymptotes of the graph of H. Figure 39 Oblique asymptote y x Table 12 Table 13 Table 14 Table 15 Figure 38 (d) As x S c-, then R (x ) S 2`; as x S c+, then R ( x ) S `. That is, the points on the graph of R are getting closer to the line x 5 c; x 5 c is a vertical asymptote. (c) As x S c–, then R (x ) S `; as x S c+, then R ( x ) S `. That is, the points on the graph of R are getting closer to the line x 5 c; x 5 c is a vertical asymptote. (b) End behavior: As x S 2`, then R ( x ) S L. That is, the points on the graph of R are getting closer to the line y 5 L; y 5 L is a horizontal asymptote. (a) End behavior: As x S `, then R ( x ) S L. That is, the points on the graph of R are getting closer to the line y 5 L; y 5 L is a horizontal asymptote. y x x 5 c y x x 5 c y 5 L y x y 5 R(x) y x y 5 L y 5 R(x) DEFINITION Horizontal and Vertical Asymptotes Let R denote a function. If, as →−∞ x or as →∞ x , the values of ( ) R x approach some fixed number L, then the line = y L is a horizontal asymptote of the graph of R. [Refer to Figures 38(a) and (b).] If, as x approaches some number c, the values ( ) →∞ R x [that is, ( ) →−∞ R x or ( ) →∞ R x ], then the line = x c is a vertical asymptote of the graph of R. [Refer to Figures 38(c) and (d).] A horizontal asymptote, when it occurs, describes the end behavior of the graph as →∞ x or as →−∞ x . The graph of a function may intersect a horizontal asymptote . A vertical asymptote, when it occurs, describes the behavior of the graph when x is close to some number c. The graph of a rational function never intersects a vertical asymptote . There is a third possibility. If, as →−∞ x or as →∞ x , the value of a rational function ( ) R x approaches a linear expression + ≠ ax b a , 0, then the line = + ≠ y ax b a , 0, is an oblique (or slant) asymptote of R. Figure 39 shows an oblique asymptote.An oblique asymptote, when it occurs, describes the end behavior of the graph. The graph of a function may intersect an oblique asymptote . Now Work PROBLEMS 27 AND 35(C)
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