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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 δ 1 > 0 such that if 0 < | x- c | < δ 1 , then f ( x ) > A/ 2. There exists δ 2 > 0 such that if 0 < | x- c | < δ 2 , then < g ( x ) < A 2 M . Then if δ = min { δ 1 ,δ 2 } , then if 0 < | x- c | < δ , 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 δ = . Then if 0 < | x | < δ , | f ( x ) /x | is 0 if x is irrational and | x | if x is rational. In either case, | f ( x ) /x | ≤ | x | < δ = . 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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solHMK14 - Section 6.3#2 Let f g be continuous on a,b...

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