Let $A,B>0$ be fixed constants. I am trying to compute the limit $$\lim_{n\to \infty}\frac{1}{n!} \int_0^A\left[\log\frac{B}{x} \right]^n~dx, $$ where $[-]$ is the floor function.
Can we interchange the limit and the integral? The integral is an improper integral because of $\log x$, so I'm not sure we can interchange. Also we have $\lim_{n\to \infty}\frac{[\log(\frac Bx)]^n}{n!}=0$ for all $x>0$ so interchanging the limit and integral seems to be awkward. Any hints for another approach?
We first observe that for any $\delta > 0$, $$\lim_{n \to \infty} \dfrac{1}{n!}\int_{\delta}^A [\ln(B/x)]^n dx = 0. $$
So we assume that $A = B$, doing the substitution, $x = Be^{-t}$ then gives the integral to be $$ \dfrac{B}{n!}\int_0^\infty [t]^ne^{-t}dt $$
Proceeding from here by breaking into limits gives the answer