Moving exclamation marks

Calculus Level 3

\[ \large \lim_{n \to \infty} \dfrac{n!}{!n}\]

The limit above has a closed form. Find this closed form.

Give your answer to 3 decimal places.

\[ \]

Notations:

  • \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

  • \(!n\) denotes the number of derangements for \(n\) different objects, \(\displaystyle !n = n! \sum_{r=0}^n \dfrac{(-1)^r}{r!} \).

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