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lim(xrarr0) (int(0)^(x)(t^(2))/(sqrt(a+t...

`lim_(xrarr0) (int_(0)^(x)(t^(2))/(sqrt(a+t))dt)/(x-sinx)=1(agt0)`. Then the value of a is

A

`1//2`

B

`1//4`

C

2

D

4

Text Solution

Verified by Experts

The correct Answer is:
D
`underset(xrarr0)(lim)(int_(0)^(x)((t^(2))/(sqrt(a+t))dt))/(x -sin x)`
`" "=underset(xrarr0)(lim)((d)/(dx)(int_(0)^(x)(t^(2))/(sqrt(a+t))dt))/(1-cosx)" (by L' Hopital's Rule)"`
`" "=underset(xrarr0)(lim)(1)/(sqrt(a+x)).(((x)/(2))/(sin.(x)/(2)))^(2).4`
`" "=(1)/(2sqrta).1.4=1 rArr a=4.`
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