Solve this following questions

Question:

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ satisfy the equation $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$ and $\mathrm{f}(\mathrm{x}) \neq 0$ for any $\mathrm{x} \in \mathrm{R}$. If Ihe function $f$ is differentiable at $x=0$ and $f^{\prime}(0)=3$, then

$\lim _{\mathrm{h} \rightarrow 0} \frac{1}{\mathrm{~h}}(f(\mathrm{~h})-1)$ is equal to_______

Solution:

If $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) \cdot f(\mathrm{y}) \& f^{\prime}(0)=3$ then

$f(x)=a^{x} \Rightarrow f^{\prime}(x)=a^{x} . \ell n a$

$\Rightarrow f^{\prime}(0)=\ell$ na $=3 \Rightarrow a=e^{3}$

$\Rightarrow f(x)=\left(e^{3}\right)^{x}=e^{3 x}$\

$\lim _{x \rightarrow 0} \frac{f(x)-1}{x}=\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-1}{3 x} \times 3\right)=1 \times 3=3$

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