Let f:RR satisfy f(x+y)=f(x) f(y) for all x,yR and f(x)0 for any xR.

Question

Let $f:\mathbb{R}	o\mathbb{R}$ satisfy $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in\mathbb{R}$ and $f(x)
e0$ for any $x\in\mathbb{R}$. If $f$ is differentiable at $x=0$ and $f'(0)=3$, then $\displaystyle\lim_{h	o0}\frac1h(f(h)-1)$ is equal to............

Answer: 3

Accepted Answer

Setting $x=y=0$: $f(0)=f(0)^2\Rightarrow f(0)=0$ or $1$; since $f
e0$, $f(0)=1$.

By definition of the derivative, $\displaystyle\lim_{h	o0}\frac{f(h)-f(0)}{h}=f'(0)=3$, i.e. $\displaystyle\lim_{h	o0}\frac{f(h)-1}{h}=3$.

Solutions · 1