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Unformatted text preview: Garcia, Ilse – Homework 4 – Due: Sep 19 2007, 3:00 am – Inst: Fonken 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 → 3 • f ( x ) g ( x ) 3 7 f ( x ) + 3 g ( x ) ‚ when lim x → 3 f ( x ) = 1 , lim x → 3 g ( x ) = 6 . Correct answer: 0 . 12 . Explanation: By properties of limits, lim x → 3 µ f ( x ) g ( x ) 3 7 f ( x ) + 3 g ( x ) ¶ = lim x → 3 ( f ( x ) g ( x ) 3) lim x → 3 (7 f ( x ) + 3 g ( x )) . But again by properties of limits, lim x → 3 ( f ( x ) g ( x ) 3) = ‡ lim x → 3 f ( x ) ·‡ lim x → 3 g ( x ) · 3 , while lim x → 3 (7 f ( x ) + 3 g ( x )) = 7 ‡ lim x → 3 f ( x ) · + 3 ‡ lim x → 3 g ( x ) · . Consequently, limit = 6 3 7 + 18 = 3 25 ≈ . 12 . 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 → 3 { f ( x ) + g ( x ) } . 1. limit = 4 2. limit = 2 3. limit = 3 4. limit does not exist correct 5. limit = 5 Explanation: From the graph it is clear that lim x → 3 { f ( x ) + g ( x ) } does not exist . (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 . Garcia, Ilse – Homework 4 – Due: Sep 19 2007, 3:00 am – Inst: Fonken 2 1. limit = 1 3 correct 2. limit = 1 3 3. limit = 3 4. limit does not exist 5. limit = 3 6. limit = 1 2 7. limit = 1 2 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+ 1 (3 / √ x ) 1 (5 / √ x ) exists, and if it does, find its value. 1. limit = 3 5 2. limit = 5 3 3. limit does not exist 4. limit = 3 5 correct 5. limit = 5 3 Explanation: After simplification and cancellation 1 (3 / √ x ) 1 (5 / √ x ) = √ x 3 √ x 5 . On the other hand, lim x → 0+ √ x = 0 , and so √ x 3 √ 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 → 2+ p 25 x 2 exists, and if it does, find its value. 1. limit = 2 2. limit = 2 √ 6 3. limit = √ 23 4. limit = 0 5. limit = √ 21 correct 6. limit does not exist 7. limit = 5 Explanation: For x near 2 the inequality 25 x 2 > holds, so f ( x ) = p 25 x 2 Garcia, Ilse – Homework 4 – Due: Sep 19 2007, 3:00 am – Inst: Fonken 3 is well defined for such x . Consequently, by Properties of Limits, the right hand limit...
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 Spring '08
 Gilbert
 Calculus

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