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2.2 Limits Involving Infinity Quick Review In Exercises 1 – 4, find f viewing window. 1. f x 2 x 3 x3 1 f x 2 –1 , and graph f, f – 1, and y = x in the same 2. f x e x f 1 x ln x Quick Review In Exercises 1 – 4, find f viewing window. –1 3. f x tan x 1 f 1 x tan x , by 3 3 2 , 2 , and graph f, f – 1, and y = x in the same 4. f x cot x 1 f 1 x cot x 0, by 1, Quick Review In Exercises 5 and 6, find the quotient q (x) and remainder r (x) when f (x) is divided by g(x). 5. f x 2 x3 3x 2 x 1, 2 qx , 3 g x 3x 3 4 x 5 5 7 2 r x 3x x 3 3 Quick Review In Exercises 5 and 6, find the quotient q (x) and remainder r (x) when f (x) is divided by g(x). 6. f x 2 x5 x3 x 1, qx 2 x 2 2 x 1, g x x 3 x 2 1 r x x 2 x 2 Quick Review In Exercises 7 – 10, write a formula for (a) f (– x) and (b) f (1/x). Simplify when possible 7. f x cos x f x cos x 1 1 f cos x x ln x 9. f x x ln x f x x 1 f x ln x x 8. f x e x f x e x 1 1 f e x x 1 10. f x x sin x x 1 f x x sin x x 1 1 1 f x sin x x x What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞ Essential Question How can limits 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 1. Use the graph and tables to find each: x 1 f x x (a) lim f x 1 x (b) lim f x 1 x (c) Identify all horizontal asymptotes. y 1 Example Sandwich Theorem Revisited The sandwich theorem also hold for limits as x → cos x 2. Find lim graphicall y and using a table of values. x x The function oscillates about the x-axis. Therefore y = 0 is the horizontal asymptote. cos x lim 0 x x Properties of Limits as x→±∞ If L, M and k are real numbers and lim f x L x 1. Sum Rule : and lim g x M , then x lim f x g x L M x The limit of the sum of two functions is the sum of their limits. 2. Difference Rule : lim f x g x L M x The limit of the difference of two functions is the difference of their limits Properties of Limits as x→±∞ 3. Product Rule: lim f x g x L M x The limit of the product of two functions is the product of their limits. 4. Constant Multiple Rule: lim k f x k L x The limit of a constant times a function is the constant times the limit of the function. 5. Quotient Rule : f x L lim , M 0 x g x M The limit of the quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. Properties of Limits as x→±∞ 6. If r and s are integers, s 0, then Power Rule : r s r s lim f x L x r s provided that L is a real number. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number. Infinite Limits as x→a If the values of a function f ( x) outgrow all positive bounds as x approaches a finite number a, we say that lim f x . If the values of f become large xa and negative, exceeding all negative bounds as x approaches a finite number a, we say that lim f x . xa Vertical Asymptote The line x a is a vertical asymptote of the graph of a function y f x if either lim f x or lim f x x a x a Example Vertical Asymptote 3. Find the vertical asymptotes of the graph of f (x) and describe the behavior of f (x) to the right and left of each vertical asymptote. 8 The vertical asymptotes are: f x 4 x 2 x = – 2 and x = 2 The value of the function approach – to the left of x = – 2 The value of the function approach + to the right of x = – 2 The value of the function approach + to the left of x = 2 The value of the function approach – to the right of x = 2 8 8 lim lim 2 2 x 2 4 x x 2 4 x 8 8 lim lim 2 2 x 2 4 x x 2 4 x End Behavior Models The function g is a a right end behavior model for f if and only if lim b a left end behavior model for x f if and only if lim x f x g x f x g x 1. 1. If one function provides both a left and right end behavior model, it is simply called an end behavior model. In general, g x an x n is an end behavior model for the polynomial function f x an x n an 1 x n 1 ... a0 , an 0 Overall, all polynomials behave like monomials. Example End Behavior Models 4. Find the end behavior model for: 3x 2 2 x 5 3x 2 is an end behavior model for the numerator. f x 2 2 4 x is an end behavior model for the denominato r. 4x 7 3x 2 3 This makes 2 an end behavior model for f x . 4x 4 3 The line y is also the horizontal asymptote of f x . 4 We can use the end behavior model of a ration function to identify any horizontal asymptote. A rational function always has a simple power function as an end behavior model. Example “Seeing” Limits as x→±∞ We can investigate the graph of y = f (x) as x → by investigating the graph of y = f (1/x) as x → 0. 1 5. Use the graph of y f to find lim f x and lim f x of x x x 1 f x x cos . x cos x 1 The graph of y f x x 1 lim f x lim f x x 0 x 1 lim f x lim f x x 0 x Pg. 66, 2.1 #1-47 odd