solHMK14 - Section 6.3 #2) Let f , g be continuous on [ a,b...

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Unformatted text preview: Section 6.3 #2) Let f , g be continuous on [ a,b ], differentiable on ( a,b ), c [ a,b ] and g ( x ) > 0 for x [ a,b ] ,x 6 = c . Let A := lim x c f , and suppose A > 0 and 0 = lim x c g . Let M > 0. There exists 1 > 0 such that if 0 < | x- c | < 1 , then f ( x ) > A/ 2. There exists 2 > 0 such that if 0 < | x- c | < 2 , then < g ( x ) < A 2 M . Then if = min { 1 , 2 } , then if 0 < | x- c | < , f ( x ) /g ( x ) > A 2 2 M A = M . Therefore, lim x c f/g = + . If A < 0, then replace f by- f and apply the above argument. So lim x c (- f ) /g = + , so lim x c f/g =- . #4) sin x is differentiable everywhere, so we must first show that f ( x ) = x 2 x Q x / Q is differentiable at 0. Let > 0 and set = . Then if 0 < | x | < , | f ( x ) /x | is 0 if x is irrational and | x | if x is rational. In either case, | f ( x ) /x | | x | < = . Hence, f is differentiable at 0, and the derivative is 0. By Theorem 6 . 3 . 1, lim x f/g exists and equals f ( a ) /g ( a ) = 0 / 1 = 0. 6 . 3 . 3 does not apply because f is not continuous, much less differentiable.is not continuous, much less differentiable....
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This note was uploaded on 01/31/2011 for the course AMS 319 taught by Professor Mark during the Fall '10 term at SUNY Stony Brook.

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solHMK14 - Section 6.3 #2) Let f , g be continuous on [ a,b...

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