HW03-solutions - le (chl528) HW03 Gilbert (57195) This...

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le (chl528) – HW03 – Gilbert – (57195) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points Suppose that f ( x ) is defned For all x in U = (5 , 6) (6 , 7) and that lim x 6 f ( x )= L. Which oF the Following statements are then true? A. if L> 0 , then f ( x ) > 0 on U ; B. if f ( x ) > 0 on U , then L 0; C. if L =0 , then f ( x ) = 0 on U . 1. B , C only 2. B only correct 3. A, C only 4. each oF A, B , C 5. A, B only 6. none oF Explanation: A. ±alse: consider the Function f ( x ) = 1 - 2 | x - 6 | . Its graph is 2 4 6 so lim x 6 f ( x ) = 1 . But on (5 , 11 2 ) and on ( 13 2 , 7) we see that f ( x ) < 0. True: iF f ( x ) > 0 on U , then on U the graph oF f always lies above the x -axis. So as x approaches 6, the point ( x, f ( x )) on the graph approaches the point (6 ,L ). Thus L 0; notice that L = 0 can occur as the graph 2 4 6 2 oF f ( x | x - 6 | shows. C. ±alse: consider the Function f ( x | x - 6 | . Its graph 2 4 6 2 shows that f ( x ) > 0 For all x ± = 6, but lim x 6 | x - 6 | . keywords: limit, defnition oF limit, properties oF limits, T/±, True/±alse 002 (part 1 oF 2) 10.0 points (i) Which oF the Following statements are true For all values oF c ? I. lim x c f ( x ) = 0 = lim x c | f ( x ) | . II. lim x c | f ( x ) | = 0 = lim x c f ( x ) = 0 . 1. I only 2. II only 3. neither I nor II
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le (chl528) – HW03 – Gilbert – (57195) 2 4. both I and II correct Explanation: 003 (part 2 of 2) 10.0 points (ii) Which of the following statements are true for all c and all L ? I. lim x c f ( x )= L = lim x c | f ( x ) | = | L | . II. lim x c | f ( x ) | = | L | = lim x c f ( x L. 1. I only correct 2. both I and II 3. neither I nor II 4. II only Explanation: 004 10.0 points A function f is deFned piecewise for all x ± = 0 by f ( x 3+ 1 2 x, x < - 2 , 5 2 x, 0 < | x | ≤ 2 , 5+ x - 1 2 x 2 , x > 2 . By Frst drawing the graph of f , determine all the values of a at which lim x a f ( x ) exists, expressing your answer in interval no- tation. 1. ( -∞ , - 2) ( - 2 , 0) (0 , ) 2. ( -∞ , - 2) ( - 2 , 0) (0 , 2) (2 , ) 3. ( -∞ , 0) (0 , ) 4. ( -∞ , 2) (2 , ) 5. ( -∞ , 0) (0 , 2) (2 , ) 6. ( -∞ , - 2) ( - 2 , ) correct 7. ( -∞ , - 2) ( - 2 , 2) (2 , ) Explanation: The graph of f is 24 - 2 - 4 2 4 - 2 - 4 and inspection shows that lim x a f ( x ) will exist only for a in ( -∞ , - 2) ( - 2 , ) .
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This note was uploaded on 02/28/2010 for the course M 52470 taught by Professor Radin during the Spring '10 term at University of Texas at Austin.

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HW03-solutions - le (chl528) HW03 Gilbert (57195) This...

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