If l i m x → ∞ a e x + b c o s x + c + d x x s i n 2 x = 3 , then the value of 272 a b d c 3 is equal to

Step-by-step solution

Using expansions, we get, l i m x → 0 a 1 + x + x 2 2 ! + x 3 3 ! + . . + b 1 - x 2 2 ! + . . . + c + d x x x - x 3 3 ! + . . . . . 2 = 3 l i m x → 0 a + b + c + a + d x + a - b 2 x 2 + a 6 x 3 + … x 3 1 - x 2 3 ! + . . . . . 2 = 3 ∵ in the denominator lowest power of x is 3 For the limit to be finite, the numerator should also have the least power of x as 3 ∴ a + b + c = 0     . . . 1 a + d = 0 . . . 2 a - b 2 = 0 . . . 3 Now, a 6 1 = 3 ⇒ a = 18 From 1 , 2 , 3 , we g