The value of lim x → 1 - ⁡ 2 π - 4 sin - 1 ⁡ x 1 - x is equal to

Step-by-step solution

We have, lim x → 1 - ⁡ π - 2 sin - 1 ⁡ x 1 - x 2 = lim x → 1 - ⁡ π - 2 sin - 1 ⁡ x 1 - x × π + 2 sin - 1 ⁡ x π + 2 sin - 1 ⁡ x × 2 = lim x → 1 - ⁡ 2 2 π 2 - sin - 1 ⁡ x 1 - x π + 2 sin - 1 ⁡ x = lim x → 1 - ⁡ 2 2 cos - 1 ⁡ x 1 - x ⋅ 1 2 π = lim θ → 0 + ⁡ 2 2 θ 2 sin ⁡ θ 2 ⋅ 1 2 π [Putting x = cos ⁡ θ ] = 4 2 2 2 π = 2 π