Just can't do this limit.. I faintly remember something similar to this appeared in first year math, but couldn't locate it.
In this question, f is continuous and bounded, so its integrals (i.e. $\int f$, $\int \int f$ ,...) will be continuous, and can be eliminated by the exp. term. I first tried to integrate by part, integrating f. That would cancel the $n^n$ term and the $(n-1)!$ term, but ended up in an n times integral of f.