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Question

If fx=1-cos xx sin x,x012 ,x=0

then at x = 0, f (x) is
(a) continuous and differentiable
(b) differentiable but not continuous
(c) continuous but not differentiable
(d) neither continuous nor differentiable

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Solution

(a) continuous and differentiable


we have,

fx=1-cos xx sin x,x012 ,x=0

fx=1-cos xx sin x,x012 ,x=0Continuity at x = 0(LHL at x = 0) = limx 0- f(x) = limh 0 f(0 - h) = limh 0 f(- h) = limh 0 1-cos (-h)(-h) sin (-h) = limh 0 1-cos hh sin h = limh 0 1-cosh limh 0 1h sin h = 1 -cos(0) . 10 sin 0 = 0


(RHL at x = 0) = limx 0+ f(x) = limh 0 f(0 + h) = limh 0 f( h) = limh 0 1-cos (h)(h) sin (h) = limh 0 1-cos hh sin h = limh 0 1-cosh limh 0 1h sin h = 1 - cos 0. 10 sin 0 = 0


Hence, f(x)is continuous at x = 0.


For differentiability at x = 0

(LHD at x = 0 ) =limx 0- f(x) - f(0)x -0 = limh0 f(0 - h) - f(0)0 - h -0 = limh0 f(- h) -12- h = limh0 1 - cos(-h)- h sin(-h) -12- h = 1h limh0 1 - cosh h sin h -limh0 12 = 12 - 0= 12

RHD at x = 0 ) =limx 0+ f(x) - f(0)x -0 = limh0 f(0 + h) - f(0)0 - h -0 = limh0 f( h) -12- h = limh0 1 - cos(h)- h sin(h) -12- h = -1hlimh0 1 - cosh h sin h -limh0 12 = 12 - 0 = 12


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