A Peculiar Infinite Limit!

$L= \lim_{n \to \infty} \left[ (n+1)! (e^{x_{n+1}} - e^{x_n})\right]$

$= \lim_{n \to \infty} \left[ (n+1)! \cdot e^{x_n}(e^{x_{n+1} - x_n} - 1)\right]$

$= \lim_{n \to \infty} \left[ (n+1)! \cdot e^{x_n}\left (e^{\frac{1}{(n+1)!}} - 1 \right)\right]$

$= \lim_{n \to \infty} e^{x_n} \times \lim_{n \to \infty} \left( \frac{e^{\frac{1}{(n+1)!}} - 1}{\frac{1}{(n+1)!}}\right)$

$= e^e \times 1 = e^e$

Therefore, $\ln(L) = \ln(e^e) = e \approx \boxed{2.718}$ .