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I am having difficulty writing this formally. I know that by L'Hospital's rule we can reduce it to $\lim_{n \to \infty} \frac{n+1}{n}$ which is a constant and hence $n = \theta (n+1)!$. But I am not sure how to write this formally.

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I am afraid that you could have been more careful.

I know that by L'Hospital's rule we can reduce it to $\lim_{n \to \infty} \frac{n+1}{n}$ which is a constant and hence $n = \theta (n+1)!$.

By L'Hospital's rule we have $$\lim_{n \to \infty} \frac{(n+1)!}{n!}=\lim_{n \to \infty} (n+1)=\infty\,, $$ which means $(n+1)! = \omega(n!)$, or what is equivalent, $n! = o(n+1)!$. Note that both $(n+1)! = \Omega(n!)$ and, what is equivalent, $n! = O(n+1)!$ are correct, which are, however, less precise.

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