Difficult limits of sequences

[Sebastian L]

Hello, I have spent a long time trying to work these limits of sequences out. I have tried to rearrange them to get them in forms that suit the standard limits, but keep on getting stuck. Any help would be great. Thank you!

 

[Deleted member 4993]

View attachment 31733View attachment 31734

Hello, I have spent a long time trying to work these limits of sequences out. I have tried to rearrange them to get them in forms that suit the standard limits, but keep on getting stuck. Any help would be great. Thank you!

As displyed, your post does NOT make sense to me. Please review it and post corrected version.

 

[BigBeachBanana]

It looks like two separate limits and they're unrelated. I'll use this thread for the first limit. Please post the second problem in another thread.
Consider the quotient rule of limits and evaluate them separately:
[math]\lim_{n \to \infty} \frac{n^{e^{-n}}}{n!} =\frac{\lim_{n \to \infty}n^{e^{-n}}} {\lim_{n \to \infty}n!}[/math]The limit on the bottom should be trivial. Please show what you can do with the top limit.