HW03.pdf - young(toy68 HW03 villafuerte altu(53780 This...

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young (toy68) – HW03 – villafuerte altu – (53780) 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 10.0points Let F be the function defined by F ( x ) = x 2 64 | x 8 | . Determine if the limit lim x 8 + F ( x ) exists, and if it does, find its value. 1. limit = 16 2. limit = 8 3. limit = 8 4. limit does not exist 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 . By properties of limits, therefore, the limit exists and lim x 8 + F ( x ) = 16 . 002 10.0points Find the value of lim x 1 f ( x ) when f ( x ) = braceleftbigg 6 x 5 , x negationslash = 1 , 2 , x = 1 , if the limit exists. 1. limit = 1 2. limit = 1 correct 3. limit = 0 4. limit = 2 5. limit = 2 6. limit does not exist Explanation: By properties of limits, lim x 1 f ( x ) = lim x 1 (6 x 5) = 6 parenleftBig lim x 1 x parenrightBig 5 , in other words, the lim x 1 f ( x ) exists and limit = 1 . 003 10.0points Determine the value of lim x 1 2 f ( x ) g ( x ) 4 f ( x ) g ( x ) when lim x 1 f ( x ) = 2 , lim x 1 g ( x ) = 1 . Correct answer: 0 . 444444. Explanation: By properties of limits lim x 1 2 f ( x ) g ( x ) = 2 lim x 1 f ( x ) lim x 1 g ( x ) = 4
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young (toy68) – HW03 – villafuerte altu – (53780) 2 while lim x 1 4 f ( x ) g ( x ) = 4 lim x 1 f ( x ) lim x 1 g ( x ) = 9 negationslash = 0 . By properties of limits again, therefore, lim x 1 2 f ( x ) g ( x ) 4 f ( x ) g ( x ) = 4 9 . 004 10.0points 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 → - 4 ( f ( x ) + g ( x )) . 1. limit = 4 2. limit = 2 3. limit does not exist correct 4. limit = 2 5. limit = 0 Explanation: From the graph it is clear that lim x → - 4 ( 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.) 005 10.0points Determine if lim x 0 braceleftbigg 4 x 12 x 2 + 3 x bracerightbigg exists, and if it does, find its value. 1. limit = 3 4 2. limit = 4 3 correct 3. limit = 4 3 4. the limit does not exist 5. limit = 3 4 Explanation: If we try to take the limit of each fraction separately, we would get the indeterminate ∞ − ∞ . We need to bring the expression to a common denominator: 4 x 12 x 2 + 3 x = 4( x + 3) 12 x ( x + 3) = 4 x x ( x + 3) = 4 x + 3 , so long as x negationslash = 0. Thus lim x 0 braceleftbigg 4 x 12 x 2 + 3 x bracerightbigg = lim x 0 4 x + 3 . But by Properties of limits, lim x 0 4 x + 3 = 4 3 . Consequently lim x 0 braceleftbigg 4 x 12 x 2 + 3 x bracerightbigg
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young (toy68) – HW03 – villafuerte altu – (53780) 3 exists and limit = 4 3 . keywords: limit, common denominator 006 10.0points Determine if lim x 3 - radicalbig 9 x 2 exists, and if it does, find its value.
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  • Fall '09
  • GOGOLEV
  • Limit, Limit of a function, lim g

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