Question

Let f : R `rightarrow` R be a differentiable function satisfying f'(3) + f'(2) = 0. Then `lim_(x rightarrow 0)((1 + f(3 + x) - f(3))/(1 + f(2 - x) - f(2)))^(1/x)` is equal to ______.

Options

• 1

• e^–1

• e

• e²

Answer 1

Let f : R `rightarrow` R be a differentiable function satisfying f'(3) + f'(2) = 0. Then `lim_(x rightarrow 0)((1 + f(3 + x) - f(3))/(1 + f(2 - x) - f(2)))^(1/x)` is equal to 1.

Explanation:

I = `lim_(x rightarrow 0)((1 + f(3 + x) - f(3))/(1 + f(2 - x) - f(2)))^(1/x)`  ...[1^∞ form]

⇒ I = e^I₁, where

I₁ = `lim_(x rightarrow 0) (((1 + f(3 + x) - f(3))/(1 + f(2 - x) - f(2))))(1/x)`

=  `lim_(x rightarrow 0)(1/x) ((f(3 + x) - f(3) - f(2 - x) + f(2))/(1 + f(2 - x) - f(2)))`  ...`(0/0 "from")`

By L Hospital Rule,

I₁ = `lim_(x rightarrow 0)((f^'(3 + x) + f^'(2 - x))/1)lim_(x rightarrow 0)(1/(1 + f(2 - x) - f(2)))`

= f'(3) + f'(2) = 0

⇒ I = e^I₁ = e⁰ = 1