Question regarding use of limits

I've been thinking about proving the product rule, and I'm not sure if I can use limits in this way to do it:

A) $\displaystyle f'(x)=\lim_{h\to \0} \frac{f(x+h)-f(x)}{h}~\Rightarrow~f(x+h)=\lim_{h\to \0} f'(x)h+f(x)$

or should it be like this:

B) $\displaystyle f'(x)=\lim_{h\to \0} \frac{f(x+h)-f(x)}{h}~\Rightarrow~\lim_{h \to \0} f(x+h)=\lim_{h\to \0} f'(x)h+f(x)$

As stated in the beginning of this post I was trying to prove the product rule, and consequently set this up:

$\displaystyle F(x)=f(x)g(x)$

$\displaystyle F'(x)=\lim_{h \to \0}\frac{F(x+h)-F(x)}{h}\Rightarrow F'(x)=\lim_{h \to \0} \frac{f(x+h)g(x+h)-f(x)g(x)}{h}$

A) feels a bit suspicious, but if it works(in the case above) it would make my task very easy, and B) seems a little pointless...

I'm feeling a bit puzzled at the moment, and any clarification would be appreciated.