Interesting observation! However, I am not sure I entirely agree. Maybe you can provide me with more details on what you mean, but from what I can tell, the pattern of n! in the denominator does not hold starting with n=4. Since we can only use L'Hopital's Rule twice, as we are limited by the numerator (the limit is not in an indeterminate form anymore after 2 L'hopital's rule applications/2 derivatives), the n! pattern does not continue after n=3. You will always stop at having n(n-1) in the denominator and can't go any further to include multiplying by (n-2), (n-3) and so on, which you would need for a complete n!. You can see this when I look at case by case in the video, for n=4 in the denominator we have 12x^2, and 12 is not 4! (4! = 24). I could be wrong, but that's what I'm seeing in this limit. Let me know your thoughts, I'm happy to discuss!
