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Question

If f(x)=2sinxsin2xx3dx, where x0, then limx0f(x) has the value:

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Solution

Consider the given expression.
f(x)=(2sinxsin2x)x3dx

On differentiating with respect to x, we get
f(x)=(2sinxsin2x)x3
limx0f(x)=limx02sinxsin2xx3
limx0f(x)=limx02sinx2sinxcosxx3
limx0f(x)=limx02sinx(1cosx)x3
limx0f(x)=limx02sinxx(1cosx)x2
limx0f(x)=limx02 sinxx[2 sin2x2]x2
limx0f(x)=limx0sinxxsin2x2x24

We know that,
limx0sinxx=1

Therefore,
limx0f(x)=1×1
limx0f(x)=1

Hence, this is the correct answer.

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