Classical Limits

An unknown function $f(x)$ can be approximated by $g_n(x)$ ,

where $g_n(x) = \frac{1^{x}+2^{x}+3^{x}+...+n^{x}}{n(1^{x-1}+2^{x-1}+3^{x-1}+...+n^{x-1})}$ .

If the $\lim_{n\to\infty}g_n(x) = f(x)$ .

What is $\prod_{i=1}^{999} f(i)$

This problem is not original.