HW03-solutions

# HW03-solutions - white(taw933 – HW03 – ben-zvi...

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Unformatted text preview: white (taw933) – HW03 – ben-zvi – (55600) 1 This print-out should have 20 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 the following statements are true without further restrictions on f ? A. Range of f need not contain 4 . B. f is defined on ( a, 5) ∪ (5 , b ) for some a < 5 < b . C. As f ( x ) approaches 4 , x approaches 5 . 1. A only 2. None of them 3. All of them 4. B only 5. A and B only correct 6. A and C only 7. B and C only 8. C only Explanation: A. True: f ( x ) need only APPROACH 4. B. True: f ( x ) need only be defined NEAR x = 5. C. Not True: f ( x ) approaches 4 AS x ap- proaches 5. keywords: True/False, LimitDefn, limit 002 (part 1 of 2) 10.0 points Which of the following statements are true for all values of c ? I. lim x → c f ( x ) = 0 = ⇒ lim x → c | f ( x ) | = 0 . II. lim x → c | f ( x ) | = 0 = ⇒ lim x → c f ( x ) = 0 . 1. II only 2. Both I and II correct 3. Neither I nor II 4. I only Explanation: If f ( x ) is close to 0, then | f ( x ) | also must be close to 0. Conversely, if | f ( x ) | is close to 0, f ( x ) must also be close to 0. Therefore Both I and II are true . 003 (part 2 of 2) 10.0 points 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. II only 2. Neither I nor II 3. I only correct 4. Both I and II Explanation: If f ( x ) is close to L , then | f ( x ) | must be close to | L | no matter what the value of L is. So I is true. But II not true for all L and c . To see that, let f ( x ) = x , c = − 2 and L = 2. Then lim x → c | f ( x ) | = lim x →- 2 | x | = 2 = | L | . white (taw933) – HW03 – ben-zvi – (55600) 2 On the other hand, lim x → c f ( x ) = lim x →- 2 x = − 2 negationslash = L . Consequently, Only I is true . 004 10.0 points Below is the graph of a function f . 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 ) = 4 correct 2. lim x → - 2 f ( x ) = 7 3. lim x → - 2 f ( x ) = 12 4. lim x → - 2 f ( x ) does not exist 5. lim x → - 2 f ( x ) = 8 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 . 005 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 →- f ( x ) exist....
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HW03-solutions - white(taw933 – HW03 – ben-zvi...

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