EQ: What are limits, and how do we find them? Finite limits as x ± Horizontal Asymptote. Example Horizontal Asymptote

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1 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 ( lim f (x), 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 (- ) ( lim f (x), we mean the limit of f as x moves x ) increasingly far to the left on the number line. Slide 2-5 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 or lim f (x) = b x NOTE: IT SAYS EITHER! A graph can cross a horizontal asymptote. It is the end behavior that makes the asymptote! Slide 2-6 Example Horizontal Asymptote Use a graph and tables to find (a) Use the limits to identify all horizontal asymptotes. f (x) = x + 1 x lim f (x) and (b) lim f (x). x [-6,6] by [-5,5] (a) lim f (x) = 1 (b) lim f (x) = 1 x (c) Horizontal asymptote at y = 1. Slide 2-7 1

2 Example Sandwich Theorem Revisited The sandwich theorem also holds for limits as x or -. cos x Find lim x graphically and using a table of values. The graph and table suggest that the function oscillates about the x-axis. cos x Thus y = 0 is the horizontal asymptote and lim = 0. x Slide 2- p. 71 Properties of Limits as x ± If L, M, and k are real numbers and lim f (x) = L and lim g(x) = M 1. Sum Rule: lim ( f (x) + g(x)) = L + M The limit of the sum of two functions is the sum of their limits. 2. Difference Rule: lim ( f (x) - g(x)) = L - M The limit of the difference of two functions is the difference of their limits. Slide 2-9 p. 71 Properties of Limits as x ± 3. Product Rule: lim ( f (x) g(x)) = L M The limit of the product of two functions is the product of their limits. 4. Constant Multiple Rule: lim (k f (x)) = k L The limit of a constant times a function is the constant times the limit of the function. f (x) L 5. Quotient Rule: lim = g(x) M Slide

3 p. 71 Properties of Limits as x ± 6. Power Rule: If r and s are integers, s 0, then provide that L r r s s lim ( f (x)) = L r s 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. Slide 2-11 p. 72 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 x a and negative, exceeding all negative bounds as x approaches a number a, we say that lim f (x) = -. x a Slide 2-12 p. 72 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 Slide

4 Example Vertical Asymptote Find the vertical asymptotes of the graph of f (x) = and describe the 4 - x 2 behavior of f (x) to the right and left of each vertical asymptote. The values of the function approach - to the left of x = -2. The values of the function approach + to the right of x = -2. The values of the function approach + to the left of x = 2. The values of the function approach - to the right of x = 2. lim 4 - x 2 = - lim = + x x 2 x 2 + lim x x 2 = + lim x x 2 = - So, the vertical asymptotes are x = -2 and x = 2. [-6,6] by [-6,6] Slide 2-14 End Behavior Models The function g is f (x) (a) a right end behavior model for f if and only if lim g(x) = 1. x f (x) (b) a left end behavior model for f if and only if lim g(x) = 1. x Slide 2-15 End Behavior Models If one function provides both a left and right end behavior model, it is simply called an end behavior model. In general, g(x) = a n x n is an end behavior model for the polynomial function f (x) = a n x n + a n-1 x n a 0, a n 0. Overall, all polynomials behave like monomials. Slide

5 Example End Behavior Models Find an end behavior model for f (x) = 3x2-2x + 5 4x Notice that 3x 2 is an end behavior model for the numerator of f, and 4x 2 is one for the denominator. This makes y = 3x 2 3 4x 2 = 4 an end behavior model for f. Slide 2-17 End Behavior Models 3 In this example, the end behavior model for f, y =, is also a horizontal 4 asymptote of the graph of f. We can use the end behavior model of a rational function to identify any horizontal asymptote. A rational function always has a simple power function as an end behavior model. Slide 2-1 Example Seeing Limits as x ± f (x) = x cos ( 1 x ) What is lim f (x)? x Slide

6 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. [ ] Use the graph of y = f [ ] The graph of y = f 1 cos x x = x is shown. lim f (x) = lim f 1 x =. x 0 + lim f (x) = lim f x x 0 [ ] [ 1 x] [ 1 ] [ 1] x x = -. to find lim f (x) and lim f (x) for f (x) = x cos x Slide 2-20 Assignment AP: p. 76 #1-, odds, 35-3, odds Regular: p. 76 #21-33 odds, all Slide