Evaluate $\lim_{x\to 0} \frac{a^x -1}{x}$ without applying L'Hopital's Rule. The questions is: Evaluate $$\lim_{x\to 0} \frac{a^x -1}{x}$$ without applying L'Hopital's Rule. Does this question fundamentally same as asking if the $\lim_{x\to 0} \frac{a^x -1}{x}$ exists? rather than straightway asking to find the limit. That means are questions (1) proving if the limit of a function exists and (2) asking what is the limit of that function, essentially same question?

set $$t=a^x-1$$ then we have $$x=\frac{1}{\ln(a)}\ln(t+1)$$ and you will get $$\frac{t}{\frac{1}{\ln(a)}\ln(t+1)}$$ can you finish?