If lim x → 0 3 + α sin x + β cos x + log e ( 1 - x ) 3 tan 2 x = 1 3 , then 2 α - β is equal to :

Step-by-step solution

Given, lim x → 0 3 + αsinx + βcosx + log e ( 1 - x ) 3 tan 2 x = 1 3 ⇒ lim x → 0 3 + α x - x 3 3 ! + … + β 1 - x 2 2 ! + x 4 4 ! … + - x - x 2 2 - x 3 3 … 3 tan 2 x = 1 3 ⇒ lim x → 0 ( 3 + β ) + ( α - 1 ) x + - 1 2 - β 2 x 2 + … . 3 x 2 × x 2 tan 2 x = 1 3 The coefficient of all the powers of x except x 2 is 0 . ⇒ β + 3 = 0 , α - 1 = 0 and - 1 2 - β 2 3 = 1 3 ⇒ β = - 3 , α = 1 ⇒ 2 α - β = 2 + 3 = 5