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Question

limx01cos(1cos2x)x4

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Solution

We have,
limx01cos(1cos2x)x4

We know that
cos2x=12sin2x

Therefore,
limx01cos(1(12sin2x))x4

limx01cos(2sin2x)x4

limx01(12sin2(sin2x))x4

limx02sin2(sin2x)x4

limx02sin2(sin2x)sin4x×sin4xx4

2limx0(sin(sin2x)(sin2x))2×sin4xx4

We know that
limx 0sinxx=1

Therefore,
2×1×1

2

Hence, this is the answer.

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