Answer
Proof
Work Step by Step
We know that definition of derivative
\[F'(c)=\lim_{x\rightarrow c}\frac{F(x)-F(c)}{x-c}\;\;\;...(1)\]
It is given that $f $ and $g$ are differentiable function
\[f'(0)=\lim_{x\rightarrow 0}\frac{f(x)-f(0)}{x-0}\]
Also it is given that $f(0)=0$
\[f'(0)=\lim_{x\rightarrow 0}\frac{f(x)}{x}\;\;\;...(2)\]
Similarly,
\[g'(0)=\lim_{x\rightarrow 0}\frac{g(x)-g(0)}{x-0}\]
Also it is given that $g(0)=0$
\[g'(0)=\lim_{x\rightarrow 0}\frac{g(x)}{x}\;\;\;...(3)\]
Using (2) and (3)
\[\Rightarrow \frac{f'(0)}{g'(0)}=\frac{\lim_{x\rightarrow 0}\frac{f(x)}{x}}{\lim_{x\rightarrow 0}\frac{g(x)}{x}}\]
\[\Rightarrow \frac{f'(0)}{g'(0)}=\lim_{x\rightarrow 0}\frac{\frac{f(x)}{x}}{\frac{g(x)}{x}}\]
\[\Rightarrow \frac{f'(0)}{g'(0)}=\lim_{x\rightarrow 0}\frac{f(x)}{g(x)}\]
Hence proven.
