Question

Let $f: R \rightarrow R$ 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 $\displaystyle\lim_{h \rightarrow 0} \frac{1}{h}(f( h )-1)$ is equal to ____.

Answer 1

If $f(x+y)=f(x) \cdot f(y) \& f^{\prime}(0)=3$ then
$f(x)=a^{x} \Rightarrow f^{\prime}(x)=a^{x} . \ell$ na
$\Rightarrow f'(0)=\ell n a=3 \Rightarrow a= e ^{3}$
$\Rightarrow f(x)=\left(e^{3}\right)^{x}=e^{3 x}$
$\displaystyle\lim _{x \rightarrow 0} \frac{f(x)-1}{x}=\displaystyle\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-1}{3 x} \times 3\right)=1 \times 3=3$