calculus - Functions whose Limit is the Factorial Function

I want to know examples of functions $f(n)$ whose limit is $n!$ Now, when I say "limit", I don't mean $$\lim_{n \to \infty}\frac{f(n)}{n!}=1$$ (I already know functions like that). I'm referring to functions that $$\lim_{n \to \infty}f(n)-n!=0$$ I am looking for functions that do not include factorials itself (that bit should be obvious, but I'm just putting it on record). I am also not looking for summation or product equations.