Derivation of the Boltzmann factor using $\ln n!\sim n\ln n-n$

Question

On this site (https://bouman.chem.georgetown.edu/S98/boltzmann/boltzmann.htm), the Boltzmann factor is derived using $\ln n!=n\ln n-n$ for $n\to\infty$, which means $$\displaystyle\lim_{n\to\infty}\dfrac{\ln n!}{n\ln n-n}=1. \tag*{(*)}$$ At some further stage, they take $\exp$ of both sides of some equation based on $(*)$ to approach the Boltzmann factor, but I doubt it can yield correct results, since $n!\ne e^{n\ln n-n}$ for $n\to\infty$, i.e. $$\displaystyle\lim_{n\to\infty}\dfrac{n!}{e^{n\ln n-n}}\ne 1.$$ Perhaps I'm missing something.