I need to calculate $$\lim_{n\rightarrow \infty} \left(\frac{n!}{n^n}\right)^{\frac{1}{n}}$$ My try:  When $n!$ is large we have $n!\approx(\frac{n}{e})^n\sqrt {2\pi n}$ (Stirling approximation) $$\lim_{n\rightarrow \infty} \left(\frac{n!}{n^n}\right)^{\frac{1}{n}}=\lim_{n\rightarrow \infty}\frac{\left((\frac{n}{e})^n\sqrt {2\pi n}\right)^{\frac{1}{n}}}{n}$$ Simplifying we get, $$\frac{1}{e}\lim_{n\rightarrow \infty} \left(\sqrt{2\pi n}\right)^{\frac{1}{n}}$$  I am stuck here. I don't know how to proceed further.
I'm trying to find  $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$   I tried couple of methods: Stolz, Squeeze, D'Alambert  Thanks!  Edit: I can't use Stirling.
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$. Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G   has subgroup of order $\operatorname{lcm}(n,m)$.   This is a statement that my lecturer mentioned in my (beginners') Abstract Algebra class. I'm not sure I understand why it's true.  What I have so far: Use the abelian group structure theorem on $\langle H, K\rangle$ (finite group generated by $H$ and $K$). Then $\langle H,K\rangle=C_{a_1}C_{a_2}\dotsm$ and $n, m |\langle H,K\rangle$. Which means it's also true that $\operatorname{lcm}(n,m)|\langle H,K\rangle$. Can I leverage this to say there's a subgroup of order $\operatorname{lcm}(n,m)$? Show that  $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$  This is an exercise I was trying to do ( not homework ) and I got stuck as follows:  Working from $A \setminus ( B \setminus C )$:  $x \in A \wedge x \notin ( B \setminus C )$ $x \in A \wedge ((x \in B \wedge x \in C) \vee (x \notin B \wedge x \in C) \vee (x \notin B \wedge x \notin C) )$  At this point I'm not sure how to proceed.  Working from the other side $ ( A \setminus B) \cup ( A \cap C ) $:  $ x \in ( A \setminus B) \cup ( A \cap C ) $  $ x \in ( A \setminus B) \vee x \in ( A \cap C ) $  $ ( x \in A \wedge x \notin B ) \vee ( x \in A \wedge x \notin C ) $  $ x \in A \wedge (x \notin B \vee x \notin C ) $  $ x \in A \wedge x \notin ( B \cap C ) $  $ A \setminus ( B \cap C ) $  I could also use some formating tips like how display logical not and how to get the lines to display properly without blank lines inbetween. :-) Why doesn't 'find' show this file? 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Sometimes it is obvious from looking at the number of answers, other times you have to follow a link to find out if it has been edited or if an answer has been added or updated.  I'm guessing this is just as applicable on other SE sites - this is the kind of thing I'm suggesting:    A couple of suggestions from the comments that might be good:   To reduce clutter the word 'ago' could be removed at the same time. The new text could be visible only to established users - ie users who are likely to be reviewing/editing and would benefit from seeing at a glance what kind of change has occurred. I exported my QGIS project with the qgis2web plugin as OL3.  Can I embed this folder to a wordpress homepage or can I import it to google maps and add this to WordPress?