Pre-Calc Homework Solutions 328

Pre-Calc Homework Solutions 328 - 328 Section 8.1 (b) Part...

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Unformatted text preview: 328 Section 8.1 (b) Part (a) shows that as the number of compoundings per year increases toward infinity, the limit of interest compounded k times per year is interest compounded continuously. 50. (a) For x lim 0, f (x) g (x) 1 1 46. Answers may vary. (a) f (x) 3x 1; g(x) lim x x 1 f (x) lim x g(x) 3x x lim x 2 3 1 3 1. (b) f (x) x 1; g(x) lim x x 1 f (x ) f (x) lim x g(x) x x2 1 lim x 2x 0 x0 g (x) 1 2 1 f (x) lim x0 g (x) 2 (c) f (x) lim x x 2; g(x) f (x) g(x) x x x 2 1 1 lim x lim x 2x 1 (b) This does not contradict L'Hpital's Rule since lim f (x) 2 and lim g (x) 1. x0 t x0 51. (a) A(t) 0 e x dx e x t e 0 t 1 lim ( e t t 47. Find c such that lim f (x) lim f (x) x0 c. 1) lim t t lim x0 9x lim x0 lim x0 lim x0 3 sin 3x 5x3 9 9 cos 3x 15x2 27 sin 3x 30x 81 cos 3x 81 30 30 x0 lim A(t) 1 et t 1 1 (b) V(t) 0 (e x)2 dx e 2x 27 10 t 0 dx Thus, c 27 . This works since lim f (x) 10 x0 c f (0), so f is continuous. 48. f (x) is defined at x form 0 . ln x x x ln x ln x 1 x 1 x 1 x2 0 1 2x t e 2 0 1 2t e 2 2 V(t) lim t A (t) 1 2 0. lim f (x) leads to the indeterminate x0 ( e 2t 1) ( e e 2t t lim t 2 1) 1 2 (1) 1 2 ln x lim x0 1 x x lim x0 lim x0 x 1 0 (c) lim V(t) t0 A (t) ( e lim 2 t0 2t t 1) 1 e lim x x0 lim e x0 x ln x e 0 lim 2 0. Extend the t0 (2e e t 2t ) Thus, f has a removable discontinuity at x definition of f by letting f (0) 1. 2 (2) 1 49. (a) The limit leads to the indeterminate form 1 . Let f (k) ln f (k) r kt . k r kt ln 1 k 1 52. (a) t ln 1 1 k r k x 0.1 0.01 0.001 f (x) 0.04542 0.00495 0.00050 0.00005 t ln 1 lim k r k t 1 k lim k r r 1 k2 k 1 k2 1 0.0001 The limit appears to be 0. (b) lim x0 1 lim k rt 1 r k rt 1 r kt k rt r kt k sin x 2x 0 1 0 L'Hpital's Rule is not applied here because the limit is A0 lim 1 k k lim A0 1 not of the form limit 1. 0 or 0 , since the denominator has A0 lim e ln f (k) k A0 e rt ...
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