This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: choi (kc24547) – HW09 – BERG – (56525) 1 This printout should have 22 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine if lim x →− 1 parenleftBig x 3 + 7 x 2 + x + 1 x 2 + 3 parenrightBig exists, and if it does, find its value. 1. limit = 5 4 2. limit does not exist 3. limit = 3 2 correct 4. limit = 6 5. limit = 5 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 →− 1 ( x 3 + 7 x 2 + x + 1) = 6 , while lim x →− 1 ( x 2 + 3) = 4 . Consequently, by Properties of limits, limit = 3 2 . 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 F ( x ) g ( x ) , B. lim x → 1 g ( x ) G ( x ) , C. lim x → 1 f ( x ) g ( x ) , are indeterminate forms? 1. A only 2. B only correct 3. none of them 4. A and B only 5. B and C only 6. C only 7. all of them 8. A and C only 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. 003 10.0 points choi (kc24547) – HW09 – BERG – (56525) 2 Determine the value of lim x → f ( x ) g ( x ) when f ( x ) = e 3 x 1 , g ( x ) = x 2 + 5 x . 1. limit = 3 2 2. limit = 2 5 3. limit = 2 3 4. limit = 3 5 correct 5. limit = 5 2 6. limit does not exist 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 → 3 e 3 x 2 x + 5 . Consequently, lim x → f ( x ) g ( x ) = 3 5 . 004 10.0 points Find the value of lim x → + 6 x ln x 2 x . 1. limit = 2 2. limit = 6 3. limit = 0 4. limit = 3 5. none of the other answers 6. limit = ∞ correct 7. limit =∞ 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, 6 x ln x 2 x = 3 ln x 2 x . Thus lim x → + 6 x ln x 2 x = 3 + ∞ , which is not an indeterminate form. In fact, since the second term is positive, we see that lim x → + 6 x ln x 2 x = ∞ . 005 10.0 points Determine if lim x → sin − 1 (2 x ) tan − 1 (3 x ) exists, and if it does, find its value....
View
Full
Document
This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas.
 Spring '10
 ZHENG
 Calculus

Click to edit the document details