Homework 03-solutions

Homework 03-solutions - Version 028 Homework 03...

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Version 028 – Homework 03 – Helleloid – (58250) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Suppose lim x 5 f ( x ) = 4 . Which of these statements are true without further restrictions on f ? A. f is defined on ( a, 5) (5 , b ) for some a < 5 < b . B. As x approaches 5 , f ( x ) approaches 4 . C. Range of f contains 4 . 1. B and C only 2. A only 3. A and B only correct 4. None of them 5. C only 6. All of them 7. A and C only 8. B only Explanation: A. True: f ( x ) needs only be defined near x = 5. B. True: definition of limit C. Not True: f ( x ) needs only AP- PROACH 4. keywords: Stewart5e, True/False, definition limit limit 002 10.0 points Below is the graph of a function f . -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 2 4 6 2 4 6 2 4 6 8 2 4 Use the graph to determine lim x → - 2 f ( x ). 1. lim x → - 2 f ( x ) = 7 2. lim x → - 2 f ( x ) = 12 3. lim x → - 2 f ( x ) does not exist 4. lim x → - 2 f ( x ) = 8 5. lim x → - 2 f ( x ) = 4 correct Explanation: From the graph it is clear the f has both a left hand limit and a right hand limit at x = 2; in addition, these limits coincide. Thus lim x → - 2 f ( x ) = 4. 003 10.0 points Below is the graph of a function f .
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Version 028 – Homework 03 – Helleloid – (58250) 2 2 4 2 4 2 4 2 4 Use the graph to determine lim x 3 f ( x ). 1. lim x 3 f ( x ) = 1 2. lim x 3 f ( x ) = 2 3. lim x 3 f ( x ) = 1 4. lim x 3 f ( x ) = 0 5. lim x 3 f ( x ) does not exist correct Explanation: From the graph it is clear that f has a left hand limit at x = 3 which is equal to 2; and a right hand limit which is equal to 0. Since the two numbers do not coincide, the limit lim x 3 f ( x ) does not exist . 004 10.0 points If f oscillates faster and faster when x ap- proaches 0 as indicated by its graph determine which, if any, of L 1 : lim x 0+ f ( x ) , L 2 : lim x 0 - f ( x ) exist. 1. both L 1 and L 2 exist 2. L 1 exists, but L 2 doesn’t 3. L 1 doesn’t exist, but L 2 does correct 4. neither L 1 nor L 2 exists Explanation: For x > 0 the graph of f oscillates but the oscillations do not get smaller and smaller as x approaches 0; so L 1 does not exist. But for x < 0, the graph oscillates and the oscillations get smaller and smaller as x approaches 0; in fact, the oscillation goes to 0 as x approaches 0, so L 2 exists. Consequently, L 1 does not exist, but L 2 does . 005 10.0 points Consider the function f ( x ) = 2 x, x < 1 x, 1 x < 3 ( x 3) 2 , x 3 . Find all the values of a for which the limit lim x a f ( x ) exists, expressing your answer in interval no- tation. 1. ( −∞ , 1) ( 1 , 3) (3 , ) correct 2. ( −∞ , 1] [3 , ) 3. ( −∞ , 3) (3 , ) 4. ( −∞ , 1) ( 1 , )
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Version 028 – Homework 03 – Helleloid – (58250) 3 5. ( −∞ , ) Explanation: The graph of f is a straight line on ( −∞ , 1), so lim x a f ( x ) exists (and = f ( a )) for all a in ( −∞ , 1).
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