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February 10, 2016 Section 1.2d Functions: Asymptotes The line y = b is a horizontal asymptote of the graph of a function if the value of the function approaches b as a limit as x approaches ±∞. Window: [-10, 20] [-10, 12] Horizontal asymptotes of Rational Functions 1) If the degree of numerator is less than the degree of the denominator the horizontal asymptote is y = 0. 2) If the degree of numerator is equal to the degree of the denominator the horizontal asymptote is y = a/b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. 3) If the degree of the numerator is greater than the degree of the denominator there is no horizontal asymptote. What is the maximum number of horizontal asymptotes a graph may have? Two 1 February 10, 2016 The line x = a is a vertical asymptote of the graph of a function if the value of the function approaches ±∞ as a limit as x approaches a from either side. Window: [-10, 20] [-10, 12] Vertical asymptotes of Rational Functions Determine the domain of the function and then evaluate each one-sided limit for numbers which are not in the domain. A vertical asymptote exists if the value of any of these limits is ±∞. What is the maximum number of vertical asymptotes a graph may have? A function may have infinitely many vertical asymptotes. 2 February 10, 2016 What is the domain of f(x)? What is the domain of g(x)? ° 0 0 0 0 -∞ -∞ ∞ ∞ -0.5 -0.5 3 February 10, 2016 What is the domain of g? What limits must be evaluated to find the vertical asymptotes. ∞ -∞ -∞ ∞ 4 February 10, 2016 Use the given limits to help determine the asymptotes of the function. Then sketch a possible graph of the function. Horizontal asymptotes: y = 2 and y = -2 Vertical asymptotes: x = -1 and x = 3 6 4 2 -6 -4 -2 2 4 6 -2 -4 -6 5