lim-x-1-n-

Question Number 131938 by Raxreedoroid last updated on 09/Feb/21

\lim_{x \to \infty} \frac{1}{\sqrt{n!}} = ?

Answered by Faetma last updated on 09/Feb/21

\begin{matrix} \lim_{n \to + \infty} n! = + \infty \\ \lim_{N \to + \infty} \sqrt{N} = + \infty \\ \lim_{N^{'} \to + \infty} \frac{1}{N^{'}} = 0^{+} \end{matrix}} \lim_{n \to + \infty} \frac{1}{\sqrt{n!}} = 0^{+}

Answered by Eyass last updated on 10/Feb/21

\forall n \in N; n! > n

\Leftrightarrow, \sqrt{n!} > \sqrt{n}

\Rightarrow n \in N^{*}, 0 < \frac{1}{\sqrt{n!}} < \frac{1}{\sqrt{n}}

\lim_{n \to + \infty} (\frac{1}{\sqrt{n}}) = 0 \Rightarrow \lim_{n \to + \infty} (\frac{1}{\sqrt{n!}}) = 0

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