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Question

Solve the given limit problem limx10x-12tcost2dtx-1sinx-1.


A

is equal to 1

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B

is equal to 12

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C

0

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D

is equal to -12

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Solution

The correct option is C

0


Explanation for the correct option:

The given expression is,limx10x-12tcost2dtx-1sinx-1 1

Step-1: Finding the value of 0x-12tcost2dt

First we solve,0x-12tcost2dt

u=t2du=2t.dtt0u=0t1-x2u=1-x22=1-x4

0x-14cosu2du=12sinu0x-14=12sinx-14-sin0=12sinx-14sin0=0

On, substituting, Equation 1 becomes

limx10x-12tcost2dtx-1sinx-1=limx112sinx-14x-1sinx-1

Step-2: Using L'hospitas rule

Now, at x=1, the numerator 12sinx-14=12sin1-14=0 and the denominator x-1sinx-1=1-1sin1-1=0.

Thus, substituting the limit x1, we get 00 indeterminate form which means limit don't exist

So we can apply the L'hospitals rule which is limx1f(x)g(x)=limx1f'(x)g'(x)

Step-3: Finding the limit

Now, applying the L'hospitals rule. we get:

limx112sinx-14x-1sinx-1=limx112×cosx-14×4x-13x-1cosx-1+sinx-1=limx12x-13cosx-14x-1cosx-1+sinx-1x-1=limx121-x2cos1-x4cosx-1+sinx-1x-1=21-13×cos1-14cos1-1+limx1sinx-1x-1=01+1[limx1sinx-1x-1=1;cos0=1]=0

So, Limit of given function exist

Hence, the correct answer is option (C).


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