If lim x → 0 a e x - b cos x + c e - x x sin x = 2 , then a + b + c is equal to ________.

Step-by-step solution

Given lim x → 0 a e x - b cos x + c e - x x sin x = 2 ⇒ lim x → 0 a 1 + x + x 2 2 ! … - b 1 - x 2 2 ! + … + c 1 - x + x 2 2 ! x sin x x x = 2 ⇒ lim x → 0 a 1 + x + x 2 2 ! … - b 1 - x 2 2 ! + … + c 1 - x + x 2 2 ! x 2 = 2       ∵ lim x → 0 sin x x = 1 For the given limit to exist numerator should be 0 ,   when  x → 0 . Then, a - b + c = 0 . . . ( i ) Applying L'Hospital's rule, we get lim x → 0 a 1 + 2 x 2