Quest Assignment 1-solutions

# Quest Assignment 1-solutions - saliyev(is4663 – Quest...

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Unformatted text preview: saliyev (is4663) – Quest Assignment 1 – rodin – (54520) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points When f, g, F and G are functions such that lim x → 1 f ( x ) = 0 , lim x → 1 g ( x ) = 0 , lim x → 1 F ( x ) = 2 , lim x → 1 G ( x ) = ∞ , which, if any, of A. lim x → 1 F ( x ) g ( x ) , B. lim x → 1 g ( x ) G ( x ) , C. lim x → 1 f ( x ) g ( x ) , are indeterminate forms? 1. B and C only 2. B only correct 3. A and C only 4. A and B only 5. C only 6. none of them 7. A only 8. all of them Explanation: A. By properties of limits lim x → 1 F ( x ) g ( x ) = 2 = 1 , so this limit is not an indeterminate form. B. Since lim x → 1 = ∞· , this limit is an indeterminate form. C. By properties of limits lim x → 1 f ( x ) g ( x ) = 0 · 0 = 0 , so this limit is not an indeterminate form. 002 10.0 points Determine if lim x →∞ x 2 ln parenleftBig x + 3 x parenrightBig exists, and if it does, find its value. 1. limit = 3 2 correct 2. limit =- 2 3 3. limit = 2 3 4. limit does not exist 5. limit = 0 6. limit =- 3 2 Explanation: Since the limit has the form ∞ × 0, it is indeterminate and L’Hospital’s Rule can be applied: lim x →∞ f ( x ) g ( x ) = lim x →∞ f ′ ( x ) g ′ ( x ) with f ( x ) = ln parenleftBig x + 3 x parenrightBig , g ( x ) = 2 x , and f ′ ( x ) = 1 x + 3- 1 x , g ′ ( x ) =- 2 x 2 . saliyev (is4663) – Quest Assignment 1 – rodin – (54520) 2 In this case, f ′ ( x ) g ′ ( x ) =- x 2 2 parenleftBig x- ( x + 3) x ( x + 3) parenrightBig = 3 2 parenleftBig x x + 3 parenrightBig-→ 3 2 as x → ∞ . Consequently, the limit exists and limit = 3 2 . 003 10.0 points Find the value of lim x → 1- cos 2 x 5 sin 2 3 x . 1. limit does not exist 2. limit = 1 15 3. limit = 1 18 4. limit = 2 45 correct 5. limit = 7 90 Explanation: Set f ( x ) = 1- cos 2 x, g ( x ) = 5 sin 2 3 x . Then f, g are differentiable functions such that lim x → f ( x ) = lim x → g ( x ) = 0 ....
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## This note was uploaded on 09/15/2011 for the course M 55565 taught by Professor Rusin during the Spring '11 term at University of Texas.

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Quest Assignment 1-solutions - saliyev(is4663 – Quest...

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