HW 09 solution

# HW 09 solution - gutierrez (ig3472) – HW09 – Neitzke...

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Unformatted text preview: gutierrez (ig3472) – HW09 – Neitzke – (56585) 1 This print-out should have 22 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine if lim x →− 2 parenleftBig x 3 + 9 x 2 + x + 2 x 2 + 1 parenrightBig exists, and if it does, find its value. 1. limit = 28 5 correct 2. limit = 24 5 3. limit = 27 4 4. limit does not exist 5. limit = 23 4 Explanation: Both the limits for the numerator and de- nominator exist and the limit of the denom- inator is not equal zero. Thus L’Hospital’s rule does not apply. Now lim x →− 2 ( x 3 + 9 x 2 + x + 2) = 28 , while lim x →− 2 ( x 2 + 1) = 5 . Consequently, by Properties of limits, limit = 28 5 . keywords: 002 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 g ( x ) G ( x ) , B. lim x → 1 f ( x ) g ( x ) , C. lim x → 1 F ( x ) g ( x ) , are indeterminate forms? 1. all of them 2. A and C only 3. C only 4. A and B only 5. A only correct 6. none of them 7. B and C only 8. B only Explanation: A. Since lim x → 1 = ∞· , this limit is an indeterminate form. B. By properties of limits lim x → 1 f ( x ) g ( x ) = 0 · 0 = 0 , so this limit is not an indeterminate form. C. By properties of limits lim x → 1 F ( x ) g ( x ) = 2 = 1 , so this limit is not an indeterminate form. gutierrez (ig3472) – HW09 – Neitzke – (56585) 2 003 10.0 points Determine the value of lim x → f ( x ) g ( x ) when f ( x ) = e 2 x- 1 , g ( x ) = x 5 + 6 x . 1. limit = 5 6 2. limit does not exist 3. limit = 1 3 correct 4. limit = 6 5 5. limit = 5 2 6. limit = 2 5 Explanation: Since f, g are differentiable functions such that lim x → f ( x ) = lim x → g ( x ) = 0 , L’Hospital’s Rule can be applied: lim x → f ( x ) g ( x ) = lim x → f ′ ( x ) g ′ ( x ) = lim x → 2 e 2 x 5 x 4 + 6 . Consequently, lim x → f ( x ) g ( x ) = 1 3 . 004 10.0 points Find the value of lim x → + 2 x- ln x 6 x . 1. limit =-∞ 2. limit = 2 3. limit = 6 4. limit = 0 5. limit = 1 3 6. none of the other answers 7. limit = ∞ correct Explanation: Let’s first check if the given limit is an indeterminate form. Now ln x is defined for x > 0 and the graph of ln x has a vertical asymptote at x = 0; in addition, lim x → + ln x =-∞ . On the other hand, 2 x- ln x 6 x = 1 3- ln x 6 x . Thus lim x → + 2 x- ln x 6 x = 1 3 + ∞ , which is not an indeterminate form. In fact, since the second term is positive, we see that lim x → + 2 x- ln x 6 x = ∞ . 005 10.0 points Determine if lim x → sin − 1 (5 x ) tan − 1 (2 x ) exists, and if it does, find its value....
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## This note was uploaded on 04/13/2010 for the course MATH 408L taught by Professor Gogolev during the Spring '09 term at University of Texas.

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HW 09 solution - gutierrez (ig3472) – HW09 – Neitzke...

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