Find limit of function with argument

Question

$\lim _{x\to 0}\left(\frac{a^x-1}{x}\right)$ I replace ${a^x-1} = t$ but dont know, what doing next.

Answer 1

This is the definition of the derivative of the function $y \mapsto a^y$ at $y=0$. Equivalently, $y \mapsto \exp(y \log(a))$, which you can differentiate using the chain rule.