$$\bar L = \displaystyle \limsup_{n\to\infty} \frac{1}{\sqrt{n}} \int_0^\infty e^{-x}\left(1+\frac{x}{n}\right)^n ~dx$$
How do you show the limit superior is finite?
I actually am relatively certain the limit itself exists:
$$L = \displaystyle \lim_{n\to\infty} \frac{1}{\sqrt{n}} \int_0^\infty e^{-x}\left(1+\frac{x}{n}\right)^n ~dx \stackrel{?}{=} \sqrt{\frac{\pi}{2}}$$
I've tried the tricks I know, and nothing has worked out.
Of course if we interchange the limsup and the integral, then we get the integrand to be $e^{-x} e^x = 1$, and so the integral itself is ∞, and if we bring the square root inside first, and then interchange the limsup and integral we get the integrand to be 0, so I am faily confident of the following: $$0 \leq \bar L \leq \infty$$