Let f : R → R satisfy the equation f ( x + y ) = f ( x ) · f ( y ) for all x , y ∈ R and f ( x ) ≠ 0 for any x ∈ R . If the function f is differentiable at x = 0 and f ' ( 0 ) = 3 , then lim h → 0 1 h f h - 1 is equal to ___ .

Question

Let f : R → R satisfy the equation f ( x + y ) = f ( x ) · f ( y ) for all x , y ∈ R and f ( x ) ≠ 0 for any x ∈ R . If the function f is differentiable at x = 0 and f ' ( 0 ) = 3 , then lim h → 0 1 h f h - 1 is equal to ___ .,

MARKS App · Accepted Answer

If f ( x + y ) = f ( x ) · f ( y ) & f ' ( 0 ) = 3 then f ( x ) = a x ⇒ f ' x = a x · l n a ⇒ f ' 0 = l n a = 3 ⇒ a = e 3 ⇒ f x = e 3 x = e 3 x lim x → 0 f ( x ) - 1 x = lim x → 0 e 3 x - 1 3 x × 3 = 1 × 3 = 3

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