section_path: Applications/Asymptotics
section_name: Asymptotics
paragraph_text: Asymptotics
  
  The number e occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers e and π appear: n ! ∼ 2 π n (n e) n. {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}.}
  
  As a consequence, e = lim n → ∞ n n ! n. {\displaystyle e=\lim _{n\to \infty }{\frac {n}{\sqrt{n!}}}.}