I know this is wrong but I cannot see why. I also cannot get to the write answer even though I did this in the past.
$$\lim_{h\to0} \frac{a^h-1}{h}=\ln(a)$$
What I did was
$$\lim_{h\to0} \frac{a^h-1}{h}\left(\frac{h}{h}\right)$$ $$\lim_{h\to0} \frac{ha^h-h}{h^2}$$ Then I used l'Hopital's rule twice: $$\lim_{h\to0} \frac{h^2(h-1)a^{h-2}}{2}=0$$
So what am I doing wrong and how to do it right?
EDTI: Ok I see my mistake.
Now how do you actually compute this limit? BTW the goal is to actually derive $a^h$ with respect to h from first principles so I cannot simply use $a^h=a^h \ln(h)$