Weird Limit + Integration Mix

$\begin{aligned} L & = \lim_{n \to \infty} \frac {\sqrt n}{e^n} \exp \left(\int_0^\infty \left \lfloor \frac n{e^x} \right \rfloor dx \right) \\ & = \lim_{n \to \infty} \frac {\sqrt n}{e^n} \exp \left(\int_0^{\ln \left(\frac n{n-1}\right)} (n-1) \ dx + \int_{\ln \left(\frac n{n-1}\right)}^{\ln \left(\frac n{n-2}\right)} (n-2) \ dx + \cdots + \int_{\ln \left(\frac n3\right)}^{\ln \left(\frac n2 \right)} 2 \ dx + \int_{\ln \left(\frac n2 \right)}^{\ln n} 1 \ dx \right) \\ & = \lim_{n \to \infty} \frac {\sqrt n}{e^n} \exp \left(\sum_{k=1}^{n-1} \int_{\ln \left(\frac n{k+1} \right)}^{\ln \left(\frac nk \right)} k \ dx \right) \\ & = \lim_{n \to \infty} \frac {\sqrt n}{e^n} \exp \left(\blue{\sum_{k=2}^n (k-1) \ln k} - \sum_{k=1}^{n-1} k \ln k \right) \\ & = \lim_{n \to \infty} \frac {\sqrt n}{e^n} \exp \left(\blue{\sum_{k=2}^n k \ln k} - \sum_{k=1}^{n-1} k \ln k - \blue{\sum_{k=2}^n \ln k} \right) \\ & = \lim_{n \to \infty} \frac {\sqrt n}{e^n} \exp \left(n \ln n - \ln (n!) \right) \\ & = \lim_{n \to \infty} \frac {\sqrt n}{e^n} \left(\frac {n^n}{\blue{n!}} \right) \quad \quad \small \blue{\text{By Stirling's formula}} \\ & = \lim_{n \to \infty} \frac {\sqrt n}{e^n} \left(\frac {n^n}{\blue{\sqrt{2\pi n}\left(\frac ne \right)^n}} \right) \\ & = \frac 1{\sqrt{2\pi}} \approx \boxed{0.399} \end{aligned}$

Reference: Stirling's formula