Let α ∈ R be such that the function f x = cos - 1 1 - x 2 sin - 1 1 - x x - x 3 , x ≠ 0 α , x = 0 is continuous at x = 0 , where x = x - x , x is the greatest integer less than or equal to x . Then :

Step-by-step solution

lim x → 0 + f x = f 0 = Lim x → 0 - x = lim x → 0 + cos - 1 1 - x 2 · sin - 1 1 - x x 1 - x 1 + x = lim x → 0 + cos - 1 1 - x 2 x · 1 · 1 · π 2 Let 1 - x 2 = cos θ = π 2 lim θ → 0 + θ 1 - cos θ = π 2 lim θ → 0 + θ 2 sin θ 2 = π 2 Now, lim x → 0 - cos - 1 1 - 1 + x 2 sin - 1 - x 1 + x - 1 + x 3 lim x → 0 - π 2 - sin - 1 x 1 + x 2 + x - x lim x → 0 - π 2 1 · 2 · sin - 1