This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Villarreal, Natalie – Homework 4 – Due: Sep 19 2007, 3:00 am – Inst: Louiza Fouli 1 This printout should have 23 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Find the value of lim x → 1 • f ( x ) g ( x ) 1 5 f ( x ) + 9 g ( x ) ‚ when lim x → 1 f ( x ) = 6 , lim x → 1 g ( x ) = 9 . Correct answer: 0 . 477477 . Explanation: By properties of limits, lim x → 1 µ f ( x ) g ( x ) 1 5 f ( x ) + 9 g ( x ) ¶ = lim x → 1 ( f ( x ) g ( x ) 1) lim x → 1 (5 f ( x ) + 9 g ( x )) . But again by properties of limits, lim x → 1 ( f ( x ) g ( x ) 1) = ‡ lim x → 1 f ( x ) ·‡ lim x → 1 g ( x ) · 1 , while lim x → 1 (5 f ( x ) + 9 g ( x )) = 5 ‡ lim x → 1 f ( x ) · + 9 ‡ lim x → 1 g ( x ) · . Consequently, limit = 54 1 30 + 81 = 53 111 ≈ . 477477 . keywords: limit, laws of limits 002 (part 1 of 1) 10 points Below are the graphs of functions f and g . 4 8 4 4 8 4 8 f : g : Use these graphs to determine lim x → 8 { f ( x ) + g ( x ) } . 1. limit = 3 2. limit = 7 3. limit = 2 correct 4. limit = 2 5. limit does not exist Explanation: From the graph it is clear that lim x → 8 { f ( x ) + g ( x ) } = 2 . (Don’t forget that for a limit to exist at a point, the left and right hand limits have to exist and coincide. So determine left and right hand limits separately and use limit laws.) keywords: limit of sum of functions, graph, limit 003 (part 1 of 1) 10 points Determine lim x → 3 n 3 x 2 3 x 1 x 3 o . Villarreal, Natalie – Homework 4 – Due: Sep 19 2007, 3:00 am – Inst: Louiza Fouli 2 1. limit = 3 2. limit = 1 3 3. limit = 1 2 4. limit = 3 5. limit = 1 3 correct 6. limit = 1 2 7. limit does not exist Explanation: After simplification we see that 3 x 2 3 x 1 x 3 = 3 x x ( x 3) = 1 x for all x 6 = 3. Thus limit = lim x → 3 1 x = 1 3 . keywords: analytic limit, difference rational functions, limit, common denominators 004 (part 1 of 1) 10 points Determine if lim x → 0+ 2 (3 / √ x ) 2 (5 / √ x ) exists, and if it does, find its value. 1. limit = 3 5 correct 2. limit = 3 5 3. limit = 5 3 4. limit = 5 3 5. limit does not exist Explanation: After simplification and cancellation 2 (3 / √ x ) 2 (5 / √ x ) = 2 √ x 3 2 √ x 5 . On the other hand, lim x → 0+ √ x = 0 , and so 2 √ x 3 2 √ x 5 = 3 5 by Properties of Limits. Consequently, the given limit exists and limit = 3 5 . keywords: analytic limit, quotient radicals, keywords: 005 (part 1 of 1) 10 points Determine if lim x → 1+ p 9 x 2 exists, and if it does, find its value. 1. limit = 2 √ 2 correct 2. limit = √ 10 3. limit does not exist 4. limit = 0 5. limit = 3 6. limit = 1 7. limit = √ 11 Explanation: For x near 1 the inequality 9 x 2 > 0 holds, so f ( x ) = p 9 x 2 is well defined for such x . Consequently, by Properties of Limits, the right hand limit lim x → 1+ p 9 x...
View
Full
Document
This note was uploaded on 04/15/2011 for the course MATH 408K taught by Professor Gualdani during the Fall '09 term at University of Texas.
 Fall '09
 Gualdani

Click to edit the document details