CLT for negative log uniforms

Let $\{X_{n}\}_{n\ge 1}$ be a sequence of independent and identically distributed random variables with probability density function $$f(x)=\begin{cases}1,&\text{if }0<x<1\\0,&\text{otherwise}.\end{cases}$$ Then, the value of the limit $$\lim_{n\to\infty}P\left(-\frac{1}{n}\sum_{i=1}^{n}\ln X_{i}\le 1+\frac{1}{\sqrt{n}}\right)$$ is equal to