3 - Nguyen (ln4328) HW #3 treisman (54540) 1 This print-out...

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Unformatted text preview: Nguyen (ln4328) HW #3 treisman (54540) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Evaluate lim x - 9 parenleftBig 8 + | x + 9 | parenrightBig . 1. limit = 9 2. limit = 11 3. limit = 7 4. limit does not exist 5. limit = 10 6. limit = 8 correct Explanation: If x > 9, then | x + 9 | = x + 9, so lim x - 9 + parenleftBig 8 + | x + 9 | parenrightBig = lim x - 9 + parenleftBig 8 + x + 9 parenrightBig = 8 . On the other hand, if x < 9, then | x + 9 | = ( x + 9), so lim x - 9- parenleftBig 8 + | x + 9 | parenrightBig = lim x - 9- parenleftBig 8 ( x + 9) parenrightBig = 8 . Since the right and left limits exist and are equal, lim x - 9 parenleftBig 8 + | x + 9 | parenrightBig exists, and limit = 8 . 002 10.0 points Determine the value of lim x 1 f ( x ) when f satisfies the inequalities 5 x f ( x ) x 3 + 2 x + 2 on [0 , 1) (1 , 2]. 1. limit = 3 2. limit does not exist 3. limit = 1 4. limit = 5 correct 5. limit = 4 6. limit = 2 Explanation: Set g ( x ) = 5 x, h ( x ) = x 3 + 2 x + 2 . Then, by properties of limits, lim x 1 g ( x ) = lim x 1 5 x = 5 , while lim x 1 h ( x ) = lim x 1 parenleftBig x 3 + 2 x + 2 parenrightBig = 1 + 2 + 2 = 5 . By the Squeeze Theorem, therefore, lim x 1 f ( x ) = 5 . To see why the Squeeze theorem applies, its a good idea to draw the graphs of g and h using, say, a graphing calculator. They look like Nguyen (ln4328) HW #3 treisman (54540) 2 g : h : (1 , 5) (not drawn to scale), so the graphs of g and h touch at the point (1 , 5) while the graph of f is sandwiched between these two graphs. Thus again we see that lim x 1 f ( x ) = 5 . 003 10.0 points If f ( x ) is defined by f ( x ) = braceleftbigg 1 , x rational , 2 , x irrational , find the value of lim x 9 f ( x ) if the limit exists. 1. limit = 2 2. limit = 0 3. limit = 18 4. limit = 9 5. limit = 1 6. limit does not exist correct Explanation: The value of f ( x ) oscillates between 1 and 2, so the problem is to decide how this affects whether lim x 9 f ( x ) exists or not, and then to find the actual value of the limit if it does exist....
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3 - Nguyen (ln4328) HW #3 treisman (54540) 1 This print-out...

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