Section 3.5 Limits at Infinity NPR 1 3x 2 f ( x) 2 x 1 Discuss “end behavior” of a function on an interval Graph: NPR 2 2 3x lim 2 x x 1 As x increases without bound f(x) approaches _______. As x decreases without bound f(x) approaches _______. NPR 3 Definition of a Horizontal Asymptote The line y=L is a horizontal asymptote of the graph of f if lim f ( x) L x OR lim f ( x) L x Note: a function can have at most 2 horizontal asymptotes NPR 4 Exploration Use a graphing utility to graph y=(2x^2 +4x-6)/(3x^2+2x-16) Describe all important features of the graph. Can you find a single viewing window that shows all these features clearly? What are the horizontal asymptotes? NPR 5 NPR 6 Recall Limits at infinity: 1. If r is a positive rational number and c is any c lim 0 real number, then x Furthermore, if x^r is defined when x<0, then lim c 0 x x r xr 2. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0. NPR 7 3. If the degree of the numerator is equal the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients. 4. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function DNE. NPR 8 Examples 2 1) lim(4 2 ) x x 2x 3 ) 2) lim( x x 1 2x 5 3) lim( 2 ) x 3 x 1 3 2 x 5 4) lim( ) 2 x 3 x 1 NPR 9 More on Horizontal Asymptotes Rational functions always have the same horizontal asymptote to the right and to the left. Functions that are NOT rational may approach different horizontal asymptotes. EX: 3x 2 3x 2 lim x 2 x2 1 and lim x NPR 2 x2 1 10 The graph: NPR 11 Limits involving Trig Functions 1) lim cos x 2) lim x sin x x x NPR 12 Infinite Limits at Infinity Examples: 1) 3 lim x 2) x x 3) 2 x2 4 x lim x x 1 lim x 3 4) NPR 2x2 4x lim x x 1 13 Asymptotes www.purplemath.com/modules/asymtote4 .htm Slant asymptotes: The graph of a rational function (having no common factors) has a slant asymptote if the degree of the numerator exceeds the degree of the denominator by 1. NPR 14 Find an equation for the slant asymptote: Use long Division: 2x2 4x y x 1 NPR 15 References Larson, Hostetler, and Edwards, Calculus of a Single Variable, Houghton Mifflin Company: 2002 NPR 16