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Unformatted text preview: suleimenov (bs26835) – test1review – rusin – (55565) 1 This printout should have 50 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. These questions are for you to practice with. You also need to work on this week’s homework assignment. Together, they cover all the topics I consider to be fair game for the exam on Tuesday. The test questions will be similar (but fewer, and NOT multiplechoice). Note that this homework is NOT required and NOT graded – it is for your practice only. Ob viously you don’t have to do all fifty problems! Have fun :) 001 0.0 points Use L’Hospital’s Rule to determine which of the inequalitites A. e x < x 2 + 100, B. 100 x < e − x , C. e 2 x > xe x + 100, holds for all large x . 1. none of them 2. all of them 3. C only correct 4. A only 5. A and B only 6. A and C only 7. B only 8. B and 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 ) 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 ) = x 2 + 100 . suleimenov (bs26835) – test1review – rusin – (55565) 2 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 , holds for all large x . B. 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 . C. TRUE: 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 ) = ∞ ....
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This note was uploaded on 09/15/2011 for the course M 55565 taught by Professor Rusin during the Spring '11 term at University of Texas.
 Spring '11
 RUSIN

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