If the function f : ℝ → ℝ defined by f x = sin ( a + 1 ) x + sin x x , x < 0 b , x = 0 x + x 2 - x x 3 2 , x > 0 is continuous on ℝ , then a + b =

Step-by-step solution

Given: f x = sin ( a + 1 ) x + sin x x , x < 0 b , x = 0 x + x 2 - x x 3 2 , x > 0 Since, f x is continuous on R Therefore, f x is continuous at x = 0 . Thus, lim x → 0 - f x = lim x → 0 + f x = f x Now, lim x → 0 - f x = f x ⇒ lim x → 0 - sin ( a + 1 ) x + sin x x = b ⇒ lim x → 0 - a + 1 sin ( a + 1 ) x a + 1 x + sin x x = b ⇒ lim x → 0 - a + 1 sin ( a + 1 ) x a + 1 x + lim x → 0 - sin x x = b ⇒ a + 1 + 1 = b ⇒ b = a + 2