Let $x$ be a real number satisfying $0<x<1$. Calculate
$$I(x)=\lim_{n\to\infty}[-\frac1n\log_eP_n(\lfloor xn\rfloor)]$$
Here, $\lfloor xn\rfloor$ denotes the largest integer not greater than the real number $xn$. And
$$P_n(k)=\frac{n!}{(n-k)!k!}p^k(1-p)^{n-k}$$
where $0<p<1$ and $k$,$n$ are positive integers satisfying $k\le n$
The hint is using Stirling's formula for $n$ is huge. I derived (may be wrong) $\lim_{n\to \infty}\lfloor xn\rfloor/n=x $ in this situation, but can not finish the work.
As you may notice, the background of $P_n$ is a probability question. It is easy to tell what that is. However I do not think that has anything to do with the limit calculation.