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
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