If f x = a + b c o s x + c s i n x x 2 ; x 0 9 ; x ≤ 0 is continuous at x = 0 , then the value of a + b 5 is

Question

If f x = a + b c o s x + c s i n x x 2 ; x > 0 9 ; x ≤ 0 is continuous at x = 0 , then the value of a + b 5 is,

MARKS App · Accepted Answer

For f x to be continuous at x = 0 l i m x → 0 f x = f 0 l i m x → 0 a + b c o s x + c s i n x x 2 = 9 l i m x → 0 a + b 1 - x 2 2 ! + x 2 4 ! - . . . . + c x - x 3 3 ! + x 5 5 ! - . . . . x 2 = 9 l i m x → 0 a + b + c x + - b 2 x 2 + - c 6 x 3 . . . . . x 2 = 9 for the above limit to satisfy a + b = 0 , c = 0 and - b 2 = 9 ⇒ b = - 18 , c = 0 , a = 1 8

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