Let f , g and h be the real valued functions defined on ℝ as f x = x x , x ≠ 0 1 , x = 0 , g x = sin x + 1 x + 1 , x ≠ - 1 1 , x = - 1 and h x = 2 x - f x , where x is the greatest integer ≤ x . Then the value of lim x → 1 g h x - 1 is

Step-by-step solution

Given, Function f x = x x , x ≠ 0 1 , x = 0 ,       g x = sin x + 1 x + 1 , x ≠ - 1 1 , x = - 1 and h x = 2 x - f x To find lim x → 1 g h x - 1 Taking L . H . L , we take x = 1 - h lim h → 0 - g ( h ( 1 - h - 1 ) ) ⇒ lim h → 0 - g 2 - h - - h - h ⇒ lim h → 0 - g ( 2 ( - 1 ) + 1 ) ⇒ lim h → 0 - g - 1 = 1 Now taking R . H . L , we take x = 1 + h lim h → 0 + g ( h ( 1 + h - 1 ) ) = lim h → 0 + g 2 h - f ( h ) = lim h → 0 + g (