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Question

limxπ2(cotx-cosx)3(π-2x)


A

116

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B

18

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C

14

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D

124

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Solution

The correct option is A

116


Explanation for the correct answer:

Step 1:

Given limit is limxπ2(cotx-cosx)3(π-2x) solving by simplifying it we get,

18limxπ2cosx(1-sinx)sinxπ2-x3

Now, replacing xπ2-h without affecting the limit as h0 so,

18limh0cosπ2-h(1-sinπ2-hsinπ2-hπ2-π2-h3=18limh0sinh(1-cosh)coshh3

Step 2:

18limh0sinh2sin2h2coshh31-cosx=2sin2x2

14limh0sinhh.sinh2h22.1cosh.14

Step 3:

Now, as we know limr0sinrr=1limr0cosr=1 so, we get

14limh0sinhh.sinh2h22.1cosh.14=14×14=116

So, limxπ2(cotx-cosx)3(π-2x)=116

Hence, the correct option is (A)


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