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Question

Evaluate:limx0[ax+a-x-2]x2


A

loga2

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B

loga

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C

0

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D

None of these

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Solution

The correct option is A

loga2


Explanation for the correct option:

Finding the value of the given limit:

Simplifying the equation to determinate form and applying the limits:

limx0[ax+a-x-2]x2=[a0+a-0-2]02=[a0+a-0-2]0=[a0+a-0-2]0=[1+1-2]0=00IndeterminateForm

Using the L' hospitality rule

limx0fxgx=limx0f'xg'x

limx0[ax+a-x-2]x2=limx0axln(a)-a-xln(a)2x[ddxax=axln(a)]=a0ln(a)-a0ln(a)2×0=ln(a)-ln(a)0=00[IndeterminateForm]

Again, using the L' hospitality rule

limx0ln(a)ax-a-x2x=ln(a)2limx0axlna--a-xln(a)1ddxax=axlna=ln(a)2limx0ax(lna)+a-x(lna)1=ln(a)2a0(lna)+a0(lna)1=ln(a)22×(lna)1=ln(a)×ln(a)=ln(a)2

Thus, limx0[ax+a-x-2]x2=(loga)2

Hence, option (A) is the correct answer.


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