Problem with differentiation as a concept. I don't understand quiet good something here, for example if we want to find the derivative of the function $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(h)}{h} $ and if we compute it from the function: $ f(x) = 12 + 7x $
We get that the derivative of $f(x)$ is equal to $$\lim_{h \to 0} \frac{7h}{h}$$
But I thought that we can't divide by zero (here we cancel 0 over 0), I'm I wrong or $\displaystyle \frac{0}{0}$ equals 1? 

The limit is defined as:

We say $\lim_{x \rightarrow c} f(x)=L$ if for all $\varepsilon>0$ there exists a $\delta>0$ such that for all $x$, if $0<|x-c|<\delta$ then $0<|f(x)-L|<\varepsilon$.

In this case, we have $c=0$, so when $x=0$ the condition $$0<|x-c|$$ is not satisfied.

Note that $$\lim_{h \rightarrow 0} \frac{7h}{h} \neq \frac{\lim_{h \rightarrow 0} 7h}{\lim_{h \rightarrow 0} h}$$ where there would be the $\frac{0}{0}$ problem you mention since we have $$\lim_{h \rightarrow 0} 7h=0=\lim_{h \rightarrow 0} h.$$
