If lim x → 0 α x e x - β log e ( 1 + x ) + γ x 2 e - x x sin 2 x = 10 ,   α ,   β ,   γ ∈ R , then the value of α + β + γ is __________.

Step-by-step solution

We have, lim x → 0 α x e x - β log e ( 1 + x ) + γ x 2 e - x x sin 2 x = 10 ⇒ lim x → 0 α x 1 + x + x 2 2 ! + . . - β x - x 2 2 + x 3 3 - . . . + γ x 2 1 - x + x 2 2 ! - . . x 3 sin 2 x x 2 = 10 ⇒ lim x → 0 α x 1 + x + x 2 2 - β x - x 2 2 + x 3 3 + γ x 2 1 - x x 3 = 10 Neglecting higher powers. lim x → 0 x ( α - β ) + x 2 α + β 2 + γ + x 3 α 2 - β 3 - γ x 3 = 10 For limit to exist α - β