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MidtermSolutions2009

MidtermSolutions2009 - Calculus I Fall 2009 Midterm Exam In...

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Calculus I - Fall 2009 Midterm Exam, October 14, 2009 In the following problems you are required to show all your work and provide the necessary explana- tions everywhere to get full credit. 1. For each of the following, decide if the limit exists. If it does, find the limit. If it does not, decide also if the limit is , -∞ , or neither. (a) lim x →- 2 x 3 - 4 x 2 + x - 1 = ( - 2) 3 - 4 ( - 2) 2 + ( - 2) - 1 = - 8 - 4 4 - 2 - 1 = - 12 1 = - 12 (b) lim x 3 x 2 + 2 x - 15 x 2 - x - 6 = lim x 3 ( x - 3)( x + 5) ( x - 3)( x + 2) = lim x 3 x + 5 x + 2 = 3 + 5 3 + 2 = 8 5 (c) lim x 3 x - 3 x - 3 = 1 2 3 Solution 1: lim x 3 x - 3 x - 3 = lim x 3 ( x - 3)( x + 3) ( x - 3)( x + 3) = lim x 3 x - 3 ( x - 3)( x + 3) = = lim x 3 1 x + 3 = 1 3 + 3 = 1 2 3 Solution 2: lim x 3 x - 3 x - 3 = lim x 3 x - 3 ( x - 3)( x + 3) = lim x 3 1 x + 3 = 1 3 + 3 = 1 2 3 (d) lim x 0 ( x 4 - 5 x ) cos x tan x = - 5 Solution: lim x 0 ( x 4 - 5 x ) cos x tan x = lim x 0 x ( x 3 - 5) cos x sin x cos x = lim x 0 x ( x 3 - 5) cos 2 x sin x = = lim x 0 ( x 3 - 5) cos 2 x · x sin x · = h lim x 0 x sin x
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