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Math151Quiz2Sols - QUIZ 2(CHAPTER 10 SOLUTIONS MATH 151...

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QUIZ 2 (CHAPTER 10) SOLUTIONS MATH 151 – SPRING 2004 – KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% 1) Find the limits. Write or - • when appropriate. If a limit does not exist, and and - • are inappropriate, write “DNE” (Does Not Exist). Indicate indeterminate forms whenever appropriate, though you don’t have to indicate signs for them. Indicate when you are applying L'H ˆo pital's Rule. a) lim x x x Æ - + - - 6 2 55 7 36 lim lim lim lim / / x L H x x x x x x x D x D x x x x x Æ- ¢ Æ- Æ- - Æ- + - - Æ - + - = - = Æ - = Ê Ë Á ˆ ¯ ˜ = + ( ) - È Î Í ˘ ˚ ˙ - ( ) = + ( ) ( ) = + = - 6 2 6 1 2 2 6 1 2 6 55 7 36 6 55 7 49 7 0 36 36 0 0 0 55 7 36 1 2 55 1 2 1 4 55 1 4 6 } ( ) - + = - = - ( ) = - 6 55 1 24 49 1 24 7 1 168
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b) lim tan sin x x x x Æ + 0 lim tan sin tan sin lim sec cos sec cos x L H x x x x x x Æ ¢ Æ + Æ + = Æ = Ê Ë Á ˆ ¯ ˜ = + = + ( ) = + ( ) = 0 0 2 2 2 0 0 0 0 0 0 0 1 1 0 0 1 1 1 } 2 c) lim sin x x x Æ + + 0 2 2 lim sin x x x Æ + + + Æ Æ = 0 2 2 2 0 Do not use L'H ˆ o pital's Rule! d) lim csc cot x x x Æ + - ( ) 0 lim csc cot lim sin cos sin lim cos sin cos sin lim sin cos lim tan tan x x x L H x x x x x x x x x x x x Æ + Æ + Æ + ¢ Æ + Æ + - ( ) = - Ê Ë Á ˆ ¯ ˜ • - • ( ) = - Æ - = - = Æ = Ê Ë Á ˆ ¯ ˜ = = = = 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 } 0
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e) lim / x x x Æ • 1 lim / x x x Æ• ( ) 1 0 Let "eventually" as y x y x y x x x x x x = = = > Æ • ( ) 1 1 1 0 / / ln ln ln ln . lim ln lim ln lim ln lim / lim x x x L H x x y x x x x x x Æ• Æ• Æ Æ• Æ• ¢ Æ• Æ• = ◊ • ( ) = Ê Ë Á ˆ ¯ ˜ = = = 1 0 1 1 1 0 0 { { } Then, lim lim lim / ln x x x x y x y e e Æ• Æ•
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