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Question

Evaluate: limx01-cos2x2x


A

2

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B

-1

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C

0

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D

does not exist

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Solution

The correct option is D

does not exist


Explanation for the correct option:

Find the value oflimx01-cos2x2x

Consider the given Equation as,

I=limx01-cos2x2xI=1-cos2020I=00[IndeterminentForm]

Using the L' hospital rule

limx0fxgx=limx0f'xg'x

I=limx01-cos2x2xI=limx0121-cos2xsin2x×22×1I=limx011-cos2xsin2x2×1I=limx0sin2x2×1-cos2x

We know that

cos2θ=2cos2θ-1orcos2θ=1-2sin2θ

Then,

I=limx0sin2x2×1-1+2sin2xI=limx0sin2x2×2sinxI=limx0sin2x2sinx

I=limx0xsin2x2xsinxI=limx0xsinxRHLI=1andLHLI=-1

Hence, the correct answer is option D.


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