If f x is a polynomial satisfying f x f 1 x = f x + f 1 x and f 2 > 1 , then lim x → 1 f x is

Step-by-step solution

As given f x . f 1 x = f x + f 1 x , f 2 , ⇒ f ( x ) f ( 1 x ) − f ( x ) − f ( 1 x ) = 0 ⇒ ( f ( x ) − 1 ) ( f ( 1 x ) − 1 ) = 1 ⇒ h ( x ) h ( 1 x ) = 1 as it f ( x ) is a polynomial,hence h ( x ) will also be a polynomial ⇒ h ( x ) = ± x n ⇒ f ( x ) = ± x n + 1 Hence f ( x ) = ± x n + 1 Now f ( 2 ) > 1 ⇒ f ( x ) = x n + 1 hence lim x → 1 f ( x ) = lim x → 1 ( x n + 1 ) = 2