Let f: R arrow 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 displaystyle limh arrow 0 (1/h)(f( h )-1) is equal to .

Question

Let f: R arrow 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 displaystyle limh arrow 0 (1/h)(f( h )-1) is equal to . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

If f(x+y)=f(x) ⋅ f(y) f prime(0)=3 then f(x)=ax ⇒ f prime(x)=ax . ℓ na ⇒ f'(0)=ℓ n a=3 ⇒ a= e 3 ⇒ f(x)=(e3)x=e3 x displaystyle lim x arrow 0 (f(x)-1/x)= displaystyle lim x arrow 0((e3 x-1/3 x) × 3)=1 × 3=3

Q. 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 h→0lim​h1​(f(h)−1) is equal to ____.

If f(x+y)=f(x)⋅f(y)&f′(0)=3 then f(x)=ax⇒f′(x)=ax.ℓ na ⇒f′(0)=ℓna=3⇒a=e3 ⇒f(x)=(e3)x=e3x x→0lim​xf(x)−1​=x→0lim​(3xe3x−1​×3)=1×3=3

Q. Let $f : R \to R$ satisfy the equation $f (x + y) = f (x) \cdot f (y)$ for all $x, y \in R$ and $f (x) \neq = 0$ for any $x \in R .$ If the function $f$ is differentiable at $x = 0$ and $f^{'} (0) = 3$ , then $h \to 0 lim \frac{1}{h} (f (h) - 1)$ is equal to ____.