HW1 - steele(pss669 HW 1 cepparo(55660 This print-out...

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steele (pss669) – HW 1 – cepparo – (55660) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points When f, g, F and G are functions such that lim x 1 f ( x ) = 0 , lim x 1 g ( x ) = 0 , lim x 1 F ( x ) = 2 , lim x 1 G ( x ) = , which, if any, of A. lim x 1 g ( x ) G ( x ) ; B. lim x 1 f ( x ) g ( x ) ; C. lim x 1 f ( x ) g ( x ) ; are NOT indeterminate forms? 1. none of them 2. C only correct 3. B only 4. A and B only 5. all of them 6. B and C only 7. A and C only 8. A only Explanation: A. Since lim x 1 = · 0 , this limit is an indeterminate form. B. Since lim x 1 f ( x ) g ( x ) = 0 0 , this limit is an indeterminate form. C. By properties of limits lim x 1 f ( x ) g ( x ) = 0 · 0 = 0 , so this limit is not an indeterminate form. 002 10.0 points Use L’Hospital’s Rule to determine which of the inequalitites A. 100 x < e - x , B. e 2 x < xe x + 100, C. e x < x 2 + 100, holds for all large x . 1. A and B only 2. none of them correct 3. B only 4. B and C only 5. A and C only 6. all of them 7. A only 8. C only Explanation: The notion of limit at infinity tells us that if lim x → ∞ f ( x ) g ( x ) = , then f ( x ) g ( x ) > 1 for all large x . But then f ( x ) > g ( x )
steele (pss669) – HW 1 – cepparo – (55660) 2 holds for all large x so long as g ( x ) > 0 for large x , allowing us to multiply through the inequality by g ( x ). Similarly, if lim x → ∞ f ( x ) g ( x ) = 0 , then f ( x ) g ( x ) < 1 for all large x , so if g ( x ) > 0 for all large x , then f ( x ) < g ( x ) for all large x . On the other hand, if lim x → ∞ f ( x ) = = lim x → ∞ g ( x ) , then we can use L’Hospital’s Rule to deter- mine lim x → ∞ f ( x ) g ( x ) . Similarly, if lim x → ∞ f ( x ) = 0 = lim x → ∞ g ( x ) , then we can use L’Hospital’s Rule to deter- mine lim x → ∞ f ( x ) g ( x ) . In this way, we can use L’Hospital’s Rule to compare the rates of growth or decay of f ( x ) and g ( x ) when x → ∞ . For the three given inequalities, therefore, we have to choose appropriate f and g and make sure that g ( x ) > 0 for all large x . A. FALSE: set f ( x ) = e - x , g ( x ) = 100 x . Then lim x → ∞ f ( x ) = 0 = lim x → ∞ g ( x ) , and by applying L’Hospital’s Rule we see that lim x → ∞ f ( x ) g ( x ) = 0 . Thus the inequality 100 x > e - x , not 100 x < e - x , holds for all large x . B. FALSE: set f ( x ) = e 2 x , g ( x ) = xe x + 100 . Then lim x → ∞ f ( x ) = = lim x → ∞ g ( x ) , and by applying L’Hospital’s Rule twice we see that lim x → ∞ f ( x ) g ( x ) = . Thus the inequality e 2 x > xe x + 100 , not e 2 x < xe x + 100 , holds for all large x . C. FALSE: set f ( x ) = e x , g ( x ) = x 2 + 100 . Then lim x → ∞ f ( x ) = = lim x → ∞ g ( x ) , and by applying L’Hospital’s Rule twice we see that lim x → ∞ f ( x ) g ( x ) = . Thus the inequality e x > x 2 + 100 , not e x < x 2 + 100 ,
steele (pss669) – HW 1 – cepparo – (55660) 3 holds for all large x .
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