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3.5 Limits Involving Infinity North Dakota Sunset What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞ …and why Limits can be used to describe the behavior of functions for numbers large in absolute value. Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line. Horizontal Asymptote The line y b is a horizontal asymptote of the graph of a function y f x if either lim f x b x or lim f x b x Example Horizontal Asymptote Use a graph and tables to find a lim f x and x c Identify all horizontal asymptotes. f x f x . b xlim x 1 x f x 1 a lim x f x 1 b xlim c Identify all horizontal asymptotes. [-6,6] by [-5,5] y 1 4 1 f x x 3 2 1 -4 1 lim 0 x x -3 -2 -1 0 1 2 3 4 -1 -2 -3 -4 As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: lim f x b x or lim f x b x lim x x x2 1 lim x x x2 x lim 1 x x This number becomes insignificant as x . There is a horizontal asymptote at 1. Find: 5 x sin x lim x x 5 x sin x lim x x x sin x lim 5 lim x x x 50 5 Infinite Limits: 4 3 1 f x x 2 1 -4 As the denominator approaches zero, the value of the fraction gets very large. -3 -2 -1 0 If the denominator is negative then the fraction is negative. 2 3 4 -1 -2 -3 -4 If the denominator is positive then the fraction is positive. 1 vertical asymptote at x=0. 1 lim x 0 x 1 lim x0 x 1 lim 2 x 0 x 1 lim 2 x 0 x The denominator is positive in both cases, so the limit is the same. 1 lim 2 x 0 x Often you can just “think through” limits. 1 lim sin x x 0 lim sin x x 0 0 p Quick Quiz You may use a graphing calculator to solve the following problems. 1. A B C D E x2 x 6 Find lim if it exists x 3 x 3 1 1 2 5 does not exist Slide 2- 12 Quick Quiz 2. A B C D E 3 x 1, Find lim f x = 5 x2 x 1 , 5 3 13 3 7 does not exist x2 x2 if it exists Slide 2- 13 Quick Quiz 2. A B C D E 3 x 1, Find lim f x = 5 x2 x 1 , 5 3 13 3 7 does not exist x2 x2 if it exists Slide 2- 14 Quick Quiz 3. Which of the following lines is a horizontal asymptote for 3x3 x 2 x 7 f x 2 x3 4 x 5 3 A y x 2 B y 0 C y D E 2 3 7 5 3 y 2 y Slide 2- 15 Quick Quiz 3. Which of the following lines is a horizontal asymptote for 3x3 x 2 x 7 f x 2 x3 4 x 5 3 A y x 2 B y 0 2 3 C y D 7 5 3 y 2 E y Slide 2- 16 Practice TEXT •p. 88, #9 – 52; •p.89, # 59, 62, 63 •p. 205, #3 – 8, 15 – 20, 21 – 33 odds, 86 – 88. • p