`lim_(x->pi/2) ((1-tan(x/2))(1-sinx))/((1+tan(x/2))((pi-2x)^3))`

Options

A. `(1)/(8)`
B. 0
C. `(1)/(32)`
D. `oo`

Answer

C

Explanation

Put `x=(pi)/(2)-h " as " x to (pi)/(2), " then " h to 0`
`therefore` Given limit becomes,
`lim_(h to 0)(1-tan((pi)/(4)-(h)/(2)))/(1+tan((pi)/(4)-(h)/(2)))*((1-cosh))/((2h)^(3))=lim_(h to 0) "tan"(h)/(2)(2"sin"^(2)(h)/(2))/(8h^(3))`
`=-lim_(h to 0)(1)/(4)*("tan"(h)/(2))/((h)/(2)xx2)xx(("sin"(h)/(2))/((h)/(2)))^(2)xx(1)/(4)`
`=lim_(x to 0)(1)/(32)*(("tan"(h)/(2))/((h)/(2)))(("sin"(h)/(2))/((h)/(2)))^(2)=(1)/(32)`