Let f : R → R be a function defined as f x = sin ( a + 1 ) x + sin 2 x 2 x , if x < 0 b , if x = 0 x + b x 3 - x b x 5 / 2 , if x > 0 If f is continuous at x = 0 , then the value of a + b is equal to :

Step-by-step solution

f ( x ) is continuous at x = 0 lim x → 0 + f x = f 0 = lim x → 0 - f x       . . . 1 f ( 0 ) = b       . . . 2 lim x → 0 - f x = lim x → 0 - sin ( a + 1 ) x 2 x + sin 2 x 2 x = a + 1 2 + 1       . . . 3 lim x → 0 + f x = lim x → 0 + x + b x 3 - x b x 5 / 2 = lim x → 0 + x + b x 3 - x b x 5 / 2 x + b x 3 + x = lim x → 0 + x x 1 + b x 2 + 1 = 1 2       . . . 4 Use ( 2 ) , ( 3 )   &   ( 4 ) in ( 1