calculus - Find $lim_{ntoinfty}frac{leftlfloor frac{n+1}{2} rightrfloor!}{n!}$

Find
$$\lim_{n\to\infty}\frac{\left\lfloor \dfrac{n+1}{2} \right\rfloor!}{n!}$$

I tried Stirling's formula but I seem to get nowhere. How should I proceed?