Evaluate: `lim_(xrarr1) (1-sqrt(X))/((cos^(-1)sqrt(x))^(2))`

Explanation

`underset(xrarr1)"lim"(1-sqrt(x))/((cos^(-1)sqrt(x))^(2 ))` Let `x=cos^(2)theta`
`=underset(thetararr0)"lim"(1-cos theta)/(theta^(2))rArr cos^(2)thetararr1`
`=underset(thetararr0)"lim"(2sin^(2)(theta)/(2))/(theta^(2))`
`=underset(thetararr0)"lim"2[(sin(theta)/(2))/(theta)]^(2)= underset(thetararr0)2[(sin(theta)/(2))/(2.(theta)/(2))]^(2)`
`=2xx(1)/(4)=(1)/(2)`