# Asymptotic relation between n! and (n+1)!

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.

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.