HW03-solutions

# HW03-solutions - husain(aih243 – HW03 – Gilbert...

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Unformatted text preview: husain (aih243) – HW03 – Gilbert – (56215) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 (part 1 of 3) 10.0 points Let F be the function defined by F ( x ) = x 2 − 25 | x − 5 | . (i) Determine lim x → 5 + F ( x ) . 1. limit = 5 2. limit does not exist 3. limit = − 5 4. limit = − 10 5. limit = 10 correct Explanation: After factorization, x 2 − 25 | x − 5 | = ( x + 5)( x − 5) | x − 5 | . But, for x > 5, | x − 5 | = x − 5 . Thus F ( x ) = x + 5 , x > 5 , in which case, by properties of limits, the right hand limit lim x → 5 + F ( x ) = 10 . 002 (part 2 of 3) 10.0 points (ii) Determine lim x → 5- F ( x ) . 1. limit = 10 2. limit = 5 3. limit does not exist 4. limit = − 10 correct 5. limit = − 5 Explanation: After factorization, x 2 − 25 | x − 5 | = ( x + 5)( x − 5) | x − 5 | . But, for x < 5, | x − 5 | = − ( x − 5) . Thus F ( x ) = − ( x + 5) , x < 5 , in which case, by properties of limits, the left hand limit lim x → 5- F ( x ) = − 10 . 003 (part 3 of 3) 10.0 points (iii) Use your results for parts (i) and (ii) to determine lim x → 5 F ( x ) . 1. limit = − 5 2. limit does not exist correct 3. limit = 5 4. limit = 10 5. limit = − 10 Explanation: By parts (i) and (ii), lim x → 5 + F ( x ) negationslash = lim x → 5- F ( x ) . husain (aih243) – HW03 – Gilbert – (56215) 2 Consequently, the two-sided limit does not exist . 004 10.0 points Determine the value of lim x → 2 4 f ( x ) g ( x ) 2 f ( x ) − 3 g ( x ) when lim x → 2 f ( x ) = 1 , lim x → 2 g ( x ) = − 4 . Correct answer: − 1 . 14286. Explanation: By properties of limits lim x → 2 4 f ( x ) g ( x ) = 4 lim x → 2 f ( x ) lim x → 2 g ( x ) = − 16 while lim x → 2 2 f ( x ) − 3 g ( x ) = 2 lim x → 2 f ( x ) − 3 lim x → 2 g ( x ) = 14 negationslash = 0 . By properties of limits again, therefore, lim x → 2 4 f ( x ) g ( x ) 2 f ( x ) − 3 g ( x ) = − 8 7 . 005 10.0 points Determine lim x → 2 braceleftBig 2 x 2 − 2 x − 1 x − 2 bracerightBig . 1. limit = 2 2. limit = 1 2 3. limit = − 1 3 4. limit = − 2 5. limit = 1 3 6. limit = − 1 2 correct 7. limit does not exist Explanation: After simplification we see that 2 x 2 − 2 x − 1 x − 2 = 2 − x x ( x − 2) = − 1 x for all x negationslash = 2. Thus limit = − lim x → 2 1 x = − 1 2 ....
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## This note was uploaded on 11/30/2010 for the course M 408c taught by Professor Mcadam during the Spring '06 term at University of Texas.

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HW03-solutions - husain(aih243 – HW03 – Gilbert...

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