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JEE Main2018MathematicsDeterminantsActual

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

Options

  1. Adoes not exist
  2. Bexists and is equal to - 2
  3. Cexists and is equal to 0
  4. Dexists and is equal to 2

Correct answer

B. exists and is equal to - 2

Step-by-step solution

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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