Evaluate this limit $ \lim_{x\to\infty}\left (\frac{1}{x}\frac{a^x – 1}{a – 1} \right)^\frac{1}{x}$

Question

Please help to evaluate this limit

$$ \lim_{x\to\infty} \left(\frac{1}{x}\frac{a^x – 1}{a – 1} \right)^\frac{1}{x},$$

where $0 \leq a$ and $a \not= 1$.

I tried to logarithm from both sides, and apply taylor series but so far without success.

Answer 1

The limit of $\left(\frac1x\right)^{\frac1x}$ equals one (by taking logs). If $a>1,$ then $(a-1)^{\frac1x} \to 1,$ while $(a^x-1)^{\frac1x} \to a,$ so the limit equals $a.$ If $a<1,$ then the limit equals $1$ (exercise).