If the function f x = cos ( sin x ) - cos x x 4 is continuous at each point in its domain and f 0 = 1 k , then k is _________.

Step-by-step solution

Given lim x → 0 cos ( sin x ) - cos x x 4 = f 0 ⇒ lim x → 0 2 sin sin x + x 2 sin x - sin x 2 x 4 = 1 k ⇒ lim x → 0 2 sin sin x + x 2 sin x - sin x 2 sin x + x 2 x - sin x 2 x 4 × sin x + x 2 x - sin x 2 = 1 k ⇒ lim x → 0 2 sin x + x 2 x x - sin x 2 x 3 = 1 k ,   ∵ lim x → 0 sin x x = 1 ⇒ 2 lim x → 0 sin x + x 2 x lim x → 0 x - sin x 2 x 3 = 1 k Apply L'Hospital rule for an indeterminate form of 0 0 , we get, ⇒ 2 × ( 1