HW01-solutions - wei (jw35975) – HW01 – kalahurka –...

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Unformatted text preview: wei (jw35975) – HW01 – kalahurka – (55230) 1 This print-out should have 24 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine lim x → braceleftBig 1 x 2 + x − 1 x bracerightBig . 1. limit = 1 2. limit = 1 2 3. limit = − 1 2 4. limit = − 1 correct 5. limit = 1 3 6. limit = − 1 3 Explanation: After simplification we see that 1 x 2 + x − 1 x = 1 − ( x + 1) x ( x + 1) = − 1 x + 1 , for all x negationslash = 0. Thus limit = lim x → − 1 x + 1 = − 1 . 002 10.0 points Determine if lim h → f ( x + h ) − f ( x ) h exists when f ( x ) = 2 x 2 + 4 x + 4, and if it does, find its value. 1. limit = 2 x + 4 2. limit does not exist 3. limit = 5 x + 4 4. limit = 4 x + 4 correct 5. limit = 3 x + 4 6. limit = 6 x + 4 Explanation: Since f ( x + h ) − f ( x ) = 2( x + h ) 2 + 4( x + h ) + 4 = 2 x 2 + (4 x + 4) h + 2 h 2 + 4 , we see that f ( x + h ) − f ( x ) h = 2 h + 4 x + 4 . On the other hand, lim h → (2 h + 4 x + 4) = 4 x + 4 . Consequently, lim h → f ( x + h ) − f ( x ) h exists when f ( x ) = 2 x 2 + 4 x + 4, and has limit = 4 x + 4 . 003 10.0 points Find the value of lim x → 5 2 x − 10 √ x − √ 5 if the limit exists. 1. limit = 6 √ 5 2. limit = 3 √ 5 3. limit = 4 √ 5 correct 4. limit does not exist wei (jw35975) – HW01 – kalahurka – (55230) 2 5. limit = 2 √ 5 6. limit = 20 Explanation: Since x − 5 = ( √ x + √ 5)( √ x − √ 5) , we can rewrite the given expression as 2( √ x + √ 5)( √ x − √ 5) √ x − √ 5 = 2( √ x + √ 5) for x negationslash = 5. Thus lim x → 5 2 x − 10 √ x − √ 5 = 4 √ 5 . 004 10.0 points Determine lim x → x − 1 x 2 ( x + 4) . 1. limit = − 1 4 2. none of the other answers 3. limit = ∞ 4. limit = −∞ correct 5. limit = 0 6. limit = 1 Explanation: Now lim x → x − 1 = − 1 . On the other hand, x 2 ( x + 4) > 0 for all small x , both positive and negative, while lim x → x 2 ( x + 4) = 0 . Consequently, limit = −∞ . keywords: evaluate limit, rational function 005 10.0 points Determine lim x →∞ 6 x 2 − 2 x + 8 2 + 7 x − 3 x 2 . 1. limit = − 2 correct 2. limit = ∞ 3. limit = 0 4. none of the other answers 5. limit = 1 Explanation: Dividing the numerator and denominator by x 2 we see that 6 x 2 − 2 x + 8 2 + 7 x − 3 x 2 = 6 − 2 x + 8 x 2 2 x 2 + 7 x − 3 . On the other hand, lim x →∞ 1 x = lim x →∞ 1 x 2 = 0 . By Properties of limits, therefore, the limit = − 2 . 006 10.0 points Determine if lim x →∞ x parenleftBig radicalbig 9 x 2 + 7 − 3 x parenrightBig exists, and if it does, find its value. 1. limit = 1 2. limit = 5 6 wei (jw35975) – HW01 – kalahurka – (55230) 3 3. limit does not exist 4. limit = 7 6 correct 5. limit = 2 3 6. limit = 4 3 Explanation: After rationalization we see that x parenleftBig radicalbig 9 x 2 + 7 − 3 x parenrightBig = x parenleftbigg 9 x 2 + 7 − 9 x 2 √ 9 x 2 + 7 + 3...
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This note was uploaded on 03/26/2012 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas.

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HW01-solutions - wei (jw35975) – HW01 – kalahurka –...

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