Lim5 2 lim 5 lim 2 5 lim 2 lim 5 2 25 x c x c x c x c

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lim[5 ( ) 2 ( )] lim 5 ( ) lim 2 ( ) 5 lim ( ) 2 lim ( ) 5( 2) 2(5) 0 x c x c x c x c x c g x f x g x f x g x f x + = + = + = + = Since the limit of the denominator is 0, the limit of the quotient does not exist. Note that lim[2 ( ) ( )] 12 0. x c f x g x = 50. lim ( ) lim ( ) 4 2 →∞ →∞ = = = x x g x g x
28 Chapter 1. Functions, Graphs, and Limits 52. x 1 0.1 0.01 0.001 0.0001 1/ 1,000(1 0.05 ) x x + 1,050.00 1,051.14 1,051.26 1,051.27 1,051.27 Thus it appears, 1/ 0 lim 1,000(1 0.05 ) 1,051.27 x x x + + = . 54. 2 2 2 2 2 2 2 2 2 2 2 6 5 2 1 5 2 1 5 2 1 6 5 lim ( ) lim ( 1) lim 6 lim 1 lim 6 lim lim 1 lim lim 6 0 1 0 0 6 t t t t t t t t t t t t t t t t t t t t t t t t t t P t t →∞ →∞ →∞ →∞ →∞ →∞ →∞ →∞ →∞ + = + + = + + + = + + + = + + + = + + = In the long run, production approaches 6,000 units. 56. 5 17 17 lim lim 5 5 0 5 n n n n n →+∞ →+∞ + = + = + = The limit tells us that as more trials are conducted, the rat’s traversal time will approach a minimum time of 5 minutes. 58. (a) 2 2 2 ( ) ( ) ( ) (400 120 ) (2 300) 100 120 3 = = + + = + P x R x C x x x x x x P x 20 100
Chapter 1. Functions, Graphs, and Limits 29 (b) The maximum profit occurs when x = 20, so the event should be announced 20 days in advance. The maximum profit is $1,300,000. (c) 2 2 400 120 ( ) 2 300 + = + x x Q x x 24 (20) 2.18 11 = Q 2 2 0 0 2 0 2 0 400 120 lim ( ) lim 2 300 lim (400 120 ) lim (2 300) 400 300 1.33 + = + + = + = x x x x x x Q x x x x x At the optimal announcement time, the revenue is more than double the advertising cost. As the announcement date gets closer to the event, the revenue gets closer to 4 3 of the advertising cost. 60. (a) 2 40(0) 50 (0) 70 20 0 1 0 10 = + = + + P The current population is 20,000. (b) 2 40(2) 50 (2) 70 59.048 2 1 2 10 = + + + P 2 40(3) 50 (3) 70 63.816 3 1 3 10 = + + + P P (3) P (2) 4.768 The population increased by 4,768 during the third year. (c) 2 2 2 40 50 10 1 lim ( ) 40 50 lim 70 1 10 40 50 lim lim 70 1 10 lim lim 70 1 1 0 0 70 70 →∞ →∞ →∞ →∞ →∞ →∞ = + + + = + + + = + + + = + = t t t t t t t t t t P t t t t t t t In the long run, the population approaches 70,000. 62. Answers will vary. The answer corresponding to each problem should include a sequence of numbers approaching the limiting value of x from the right and left, along with the corresponding values of ( ). f x 64. (a) The growth rate doubles from about 0.5 generation/hr to 1.0 generations/hr between about 10 ° C and 15 ° C. It also doubles from about 0.75 generation/hr to 1.5 generations/hr between about 12 ° C and 20 ° C. (b) The growth rate is constant for 25 < T < 45. (c) The growth rate begins to decrease at about 45 ° C, then drops rapidly. It appears that lim ( ) 0. →50 = T R T (d) Writing exercise; answers will vary. 1.6 One-Sided Limits and Continuity 2. As x approaches 2 from the left, the curve approaches the point (2, 4) so 2 lim ( ) 4 x f x = . From the right the curve approaches the point (2, 2) so 2 lim ( ) 2 x f x + = . Since the one-sided limits at 2 x = are not equal, 2 lim ( ) x f x does not exist.
30 Chapter 1. Functions, Graphs, and Limits 4. As x approaches 2 from the left, the curve approaches the point (2, 2) so 2 lim ( ) 2 x f x = . From the right the curve assumes larger and larger values as it nears 2 so 2 lim ( ) x f x + = +∞ . Since the one-sided limits at 2 x = are not equal, 2 lim ( ) x f x does not exist.

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