If the function f x , defined as f x = a 1 - x sin ⁡ x + b cos ⁡ x + 5 x 2 : x ≠ 0 3 : x = 0 is continuous at x = 0 , then the value of b 4 + a 5 + a is equal to

Step-by-step solution

Since, f x is continuous at x = 0 so, at x = 0 both left hand and right hand limits must exist and both must be equal to 3 . Now, l i m x → 0 a 1 - x sin ⁡ x + b cos ⁡ x + 5 x 2 = f 0 ⇒ lim x → 0 a - a x x - x 3 3 ! + . . . . + b 1 - x 2 2 ! + . . . . + 5 x 2 = f 0 ⇒ l i m x → 0 a + b + 5 + - a - b 2 x 2 + . . . x 2 = 3 If l i m x → 0 f x = 3 exists, then a + b + 5 = 0 and - a - b 2 = 3 ⇒ a = - 1 and b = - 4