385 3.5 Rational Functions: Graphs, Applications, and Models Asymptotes The preceding examples suggest the following definitions of vertical and horizontal asymptotes. LOOKING AHEAD TO CALCULUS The rational function ƒ1x2 = 2 x + 1 in Example 2 has a vertical asymptote at x = -1. In calculus, the behavior of the graph of this function for values close to -1 is described using one- sided limits. As x approaches -1 from the left, the function values decrease without bound. This is written lim xS-1− ƒ1x2 = -∞. As x approaches -1 from the right, the function values increase without bound. This is written lim xS-1+ ƒ1x2 = ∞. Asymptotes Let p1x2 and q1x2 define polynomial functions. Consider the rational function ƒ1x2 = p1x2 q1x2 , written in lowest terms, and real numbers a and b. 1. If ƒ1x2 S∞ as x Sa, then the line x = a is a vertical asymptote. 2. If ƒ1x2 Sb as x S∞, then the line y = b is a horizontal asymptote. Locating asymptotes is important when graphing rational functions. • We find vertical asymptotes by determining the values of x that make the denominator equal to 0. • We find horizontal asymptotes (and, in some cases, oblique asymptotes) by considering what happens to ƒ1x2 as x S∞. These asymptotes determine the end behavior of the graph. Determining Asymptotes To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures. 1. Vertical Asymptotes Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x =a is a vertical asymptote. 2. Other Asymptotes Determine any other asymptotes by considering three possibilities: (a) If the numerator has lesser degree than the denominator, then there is a horizontal asymptote y =0 (the x-axis). (b) If the numerator and denominator have the same degree, and the function is of the form ƒ1x2 = an x n + g+ a 0 bn x n + g+ b 0 , where an, bn ≠0, then the horizontal asymptote has equation y = an bn . (c) If the numerator is of degree exactly one more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote. NOTE The graph of a rational function may have more than one vertical asymptote, or it may have none at all. The graph cannot intersect any vertical asymptote. There can be at most one other (nonvertical) asymptote, and the graph may intersect that asymptote. (See Example 7.)
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