How to evaluate $\lim\limits_{h \to 0} \frac {3^h-1} {h}=\ln3$? How is
$$\lim_{h \to 0} \frac {3^h-1} {h}=\ln3$$
evaluated?

There are at least two ways of doing this: Either you can use de l'Hôpital's rule, and as I pointed out in the comments the third example on Wikipedia gives the details.
I think a better way of doing this (and Jonas seems to agree, as I saw after posting) is to write $f(h) = 3^{h} = e^{\log{3}\cdot h}$ and write the limit as $$\lim_{h \to 0} \frac{f(h) - f(0)}{h}$$ and recall the definition of a derivative. What comes out is $f'(0) = \log{3}$.
