Question

`lim_(x rightarrow π/2) ([1 - tan (x/2)] (1 - sin x))/([1 + tan (x/2)] (π - 2x)^3` is ______.

पर्याय

• `1/8`

• 0

• `1/32`

• ∞

Answer 1

`lim_(x rightarrow π/2) ([1 - tan (x/2)] (1 - sin x))/([1 + tan (x/2)] (π - 2x)^3` is `underlinebb(1/32)`.

Explanation:

Given, `lim_(x rightarrow π/2) ((1 - tan  x/2) (1 - sin x))/((1 + tan  x/2) (π - 2x)^3`

Let x = `π/2 - h` as `x rightarrow π/2, h rightarrow 0`

= `lim_(h rightarrow 0) (1 - tan (π/4 - h/2))/(1 + tan(π/4 - h/2)) (1 - cos h)/(2h)^3`  ...`[∵ tan(π/4 - x) = (1 - tan x)/(1 + tan x)]`

= `lim_(h rightarrow 0) tan  h/2 . (2 sin^2  h/2)/(8h^3)`  

= `lim_(h rightarrow 0) 1/4 . (tan  h/2)/(h/2 xx 2) xx ((sin  h/2)/(h/2))^2 xx 1/4`

= `1/32`