lim-x-x-x-x-1-x-

Question Number 142329 by HarshSahu last updated on 30/May/21

{l i m}_{x \to \infty} {(\frac{x!}{x^{x}})}^{\frac{1}{x}}

Answered by Dwaipayan Shikari last updated on 30/May/21

{(\frac{x!}{x^{x}})}^{\frac{1}{x}} = y

\frac{1}{x} l o g (x! / x^{x}) = l o g (y)

\lim_{x \to \infty} \frac{1}{x} \underset{r = 1}{\sum^{x}} l o g (\frac{r}{x}) = l o g (y)

\int_{0}^{1} l o g (t) d t = l o g (y) \Rightarrow - 1 = l o g (y) \Rightarrow y = \frac{1}{e}

Answered by MJS_new last updated on 30/May/21

{(\frac{x!}{x^{x}})}^{1 / x} = \frac{{(x!)}^{1 / x}}{x} = \frac{{({(\frac{x}{e})}^{x} \sqrt{2 π x})}^{1 / x}}{x} = \frac{\frac{x}{e} \sqrt[2 x]{2 π x}}{x} =

= \frac{\sqrt[2 x]{2 π x}}{e}

\lim_{x \to \infty} \frac{\sqrt[2 x]{2 π x}}{e} = \frac{1}{e}

Commented by greg_ed last updated on 31/May/21

ok!

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