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If f(x) is the integral of (2 sin x-sin ...

If f(x) is the integral of `(2 sin x-sin 2 x)/(x^(3)), "where x" ne 0, "then find" lim_(x rarr 0) f'(x)`.

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The correct Answer is:
1
Given, `f(x)=int((2 sin x - sin 2 x)/(x^(3)))dx`
On differentiation w.r.t.x, we get `f'(x)=(2sin x - sin 2x)/(x^(3))=(2 sin x)/(x)((1-cos x)/(x^(2)))`
`underset(x rarr 0)("lim") f'(x)=underset(x rarr 0)("lim 2")((sin x)/(x))(("2 sin"^(2)(x)/(2))/(x^(2)))`
`=4*1*underset(x rarr 0)("lim")[("sin"^(2)(x)/(2))/(4 xx ((x)/(2))^(2))]=1`
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