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 ________.

Hint:

Functions satisfying $f(x+y)=f(x)f(y)$ are exponential functions. Their derivative at zero is simply the natural log of the base ($\ln a$).

Answer 1

Correct Answer: 3

Step 1: For $f(x+y) = f(x)f(y)$, the standard form is $f(x) = a^x$.
Step 2: Find $f(0)$: $f(0+0) = f(0)f(0) \implies f(0) = f(0)^2$. Since $f(x) \neq 0$, $f(0) = 1$.
Step 3: The given limit is the definition of the derivative at $x=0$: $f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0} \frac{f(h) - 1}{h}$.
Step 4: Since $f'(0) = 3$, the value of the limit is directly 3.