The value of `lim_(xto1^(-)) (1-sqrt(x))/((cos^(-1)x)^(2))` is

Options

A. `1//6`
B. `-1//3`
C. `1//2`
D. 1

Answer

D

Explanation

We have
`underset(xto1)lim(1-sqrt(x))/((cos^(-1)x)^(2))=underset(xto1)lim((1-sqrt(x))(1+sqrt(x)))/((cos^(-1)x)^(2)(1+sqrt(x)))`
`=underset(xto1)lim(1-x)/((cos^(-1)x)^(2)(1+sqrt(x)))`
`=underset(thetato0)lim(1-costheta)/((1+sqrt(costheta)))`, where `x=costheta`
`[because xto1" or "costhetato1" or "thetato0]`
`underset(thetato0)lim(1-costheta)/(theta^(2))(1)/((1+sqrt(costheta)))`
`=underset(thetato0)lim(2"sin"^(2)(theta)/(2))/(4(theta^(2))/(4))((1)/(1+sqrt(costheta)))`
`=(1)/(2)underset(thetato0)lim(("sin"(theta)/(2))/((theta)/(4)))^(2)(1)/((1+sqrt(costheta)))`
`=(1)/(2)(1)^(2)(1)/((1+1))=(1)/(4)`