If f x = cos x x 1 2 sin x x 2 2 x tan x x 1 , then lim x → 0 ⁡ f ′ x x

Question

If f x = cos x x 1 2 sin x x 2 2 x tan x x 1 , then lim x → 0 ⁡ f ′ x x, does not exist, exists and is equal to - 2, exists and is equal to 0, exists and is equal to 2

MARKS App · Accepted Answer

Given f x = cos x x 1 2 sin x x 2 2 x tan x x 1 = cos x x 2 - 2 x 2 - 2 sin x x - x + tan x 2 x 2 - x 2 = - x 2 cos x + tan x . x 2 = x 2 tan x - cos x ⇒ Lim x → 0 f ' x x = Lim x → 0 2 x tan x - cos x + x 2 sec 2 x + sin x x = Lim x → 0 2 tan x - cos x + x sec 2 x + sin x = - 2

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