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sol7 - Problem Set 7 Calculus 1 Abdullah Khalid Hassan...

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Problem Set 7 Calculus 1 Abdullah Khalid, Hassan Bukhari January 3, 2010 Announcement: I am available all day Tuesday to ask questions. Please feel free to drop by in the office any time. 1. Apply the Mean Value Theorem to the function f on the interval ( - b, b ) to find a point c such that f 0 ( c ) = f ( b ) - f ( - b ) 2 b Since f is odd, f ( b ) = - f ( - b ). Hence f 0 ( c ) = f ( b ) /b . 2. (a) Applying L’Hospital’s rule we get lim x →∞ ln ln x x = lim x →∞ 1 x ln x = 0 (b) The limit does not exist. A very intuitive reason why the limit does not exist is because sin oscillates between -1 and 1 where as ln x approaches as x approaches . (c) We know that lim x 0 + x ln x = lim x 0 + ln x x - 1 = lim x 0 + - x 2 x = lim x 0 + - x = 0 So we can rewrite lim x 0 + sin x ln x = lim x 0 + sin x x x ln x = lim x 0 + sin x x lim x →∞ x ln x = 0 1
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(d) We want to find lim x →∞ f ( x ) where f ( x ) = x 1 /x . Let us first compute the limit of ln f ( x ). lim x →∞ ln f ( x ) = lim x →∞ ln x x = 0 where the last equality follows by a simple application of L’Hospital’s rule.
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