Let's say that I wanted to prove the following problem which is similar to a homework problem. That is, the numerator and denominator are different, but similar enough that you posting an answer would likely give everything away.
$$\lim_{n\to\infty} \frac{6n^4}{9\left \lfloor{\ln n}\right \rfloor!} = 0 $$
Using L'Hopitals rule does not help here (well, maybe), and factoring does not help either.
I am not sure where I could start. I want to solve this myself so I can solve my homework problem.
Is there anything obvious that jumps off the bat about this problem that I should be seeing? I come across stuff similar to this all of the time but never know where to start and I feel like I just get lucky after enough guesses, so I want to learn how to approach the problem.
Edit:
If it helps, I actually do not care that the limit is 0. I just care about which one grows faster as $n$ gets large and being able to prove that.
Consider what happens when you go from $n$ to $en$ in the numerator and denominator.