HW5-solutions - wei (jw35975) HW5 milburn (54685) 1 This...

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Unformatted text preview: wei (jw35975) HW5 milburn (54685) 1 This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine lim x 1 braceleftBig 1 x 2 x 1 x 1 bracerightBig . 1. limit = 1 correct 2. limit = 1 2 3. limit does not exist 4. limit = 1 3 5. limit = 1 6. limit = 1 3 7. limit = 1 2 Explanation: After simplification we see that 1 x 2 x 1 x 1 = 1 x x ( x 1) = 1 x for all x negationslash = 1. Thus limit = lim x 1 1 x = 1 . 002 (part 1 of 3) 10.0 points Let F be the function defined by F ( x ) = x 2 64 | x 8 | . (i) Determine lim x 8 + F ( x ) . 1. limit = 16 2. limit does not exist 3. limit = 8 4. limit = 8 5. limit = 16 correct Explanation: After factorization, x 2 64 | x 8 | = ( x + 8)( x 8) | x 8 | . But, for x > 8, | x 8 | = x 8 . Thus F ( x ) = x + 8 , x > 8 , in which case, by properties of limits, the right hand limit lim x 8 + F ( x ) = 16 . 003 (part 2 of 3) 10.0 points (ii) Determine lim x 8- F ( x ) . 1. limit does not exist 2. limit = 16 correct 3. limit = 8 4. limit = 16 5. limit = 8 Explanation: After factorization, x 2 64 | x 8 | = ( x + 8)( x 8) | x 8 | . But, for x < 8, | x 8 | = ( x 8) . wei (jw35975) HW5 milburn (54685) 2 Thus F ( x ) = ( x + 8) , x < 8 , in which case, by properties of limits, the left hand limit lim x 8- F ( x ) = 16 . 004 (part 3 of 3) 10.0 points (iii) Use your results for parts (i) and (ii) to determine lim x 8 F ( x ) . 1. limit = 16 2. limit = 8 3. limit = 8 4. limit = 16 5. limit does not exist correct Explanation: By parts (i) and (ii), lim x 8 + F ( x ) negationslash = lim x 8- F ( x ) . Consequently, the two-sided limit does not exist . 005 10.0 points Determine lim x 1 x 1 x + 3 2 . 1. limit = 2 2. limit = 4 correct 3. limit = 1 4 4. limit doesnt exist 5. limit = 1 2 Explanation: After rationalizing the denominator we see that 1 x + 3 2 = x + 3 + 2 ( x + 3) 4 = x + 3 + 2 x 1 . Thus x 1 x + 3 2 = x + 3 + 2 for all x negationslash = 1. Consequently, limit = lim x 1 ( x + 3 + 2) = 4 . 006 10.0 points Determine if lim h f (1 + h ) f (1) h exists when f ( x ) = x 2 + 5 x , and if it does, find its value. 1. limit = 7 correct 2. limit = 11 3. limit does not exist 4. limit = 8 5. limit = 10 6. limit = 9 Explanation: Since f (1 + h ) f (1) = (6 + 7 h + h 2 ) 6 , we see that f (1 + h ) f (1) h = h 2 + 7 h h = h + 7 . wei (jw35975) HW5 milburn (54685) 3 On the other hand, lim h ( h + 7) = 7 , by Properties of Limits. Consequently, the given limit exists, and limit = 7 ....
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HW5-solutions - wei (jw35975) HW5 milburn (54685) 1 This...

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