`lim_(xrarr(pi)/(2)) ((1-sinx)(8x^(3)-pi^(3))(cosx))/(pi-2x)^(4)`

Options

A. `-(pi^(2))/(16)`
B. `(3pi^(2))/(16)`
C. `(pi^(2))/(16)`
D. `-(3pi^(2))/(16)`

Answer

D

Explanation

`underset(xrarr(pi)/(2))(lim)((1-sinx)(8x^(3)-pi^(3))cosx)/(pi-2x)^(4)`
`=underset(xrarr(pi)/(2))(lim)((1-cos((pi)/(2)-x))(2x-pi)(4x^(2)+pi^(2)+2pix)sin((pi)/(2)-x))/(16((pi)/(2)-x)^(4))`
`underset(hrarr0)(lim)((1-cos h)(-2h)sin h)/(16h^(2))underset(xrarr(pi)/(2))(lim)(4x^(2)+pi^(2)+2pix)`
`=underset(hrarr0)(lim)((1-cos h)(-2h sin h))/(16h^(4))underset(xrarr(pi)/(2))(lim)(4x^(2)+pi^(2)+2pix)`
`=-underset(hrarr0)(lim)((1-cos h))/(8h^(4))(4((pi)/(2))^(2)+pi^(2)+2pi((pi)/(2)))`
`=-(3pi^(2))underset(hrarr0)(lim)(2sin^(2).(h)/(2))/(32(h^(2))/(4))`
`=-(3pi^(2))/(16)`