I got this problem.
Prove that $\lim_{x\to 0}\dfrac{a^x-1}{x}= \ln(a)$ by using the definition of limit and by the fact $\lim_{x\to 0} \dfrac{x}{\log_{a}(1+x)}=\ln(a)$ and without using any other theorem, and of course without using L'Hospital rule.
I got stuck, I don't know which $\delta$ to choose.
Any hints will be appreciated.