I tried to do a change of variable with $t=\frac{1}{n}$ in order to use Taylor expansion but the factor $n!$ becomes a fractional number (during the lesson it was define only for $\mathbb{Z}$ set number). Another attempt I made was to write the limit as $\lim_{n \to \infty} \sqrt[n]{\frac{n!}{n^n}} \rightarrow e^{\frac{1}{n}log{\frac{n!}{n^n}}}$ but it doesn't work. After that I tried to use the Stirling formula and yes... it works! But I would find something that doesn't use this formula. Some advice?
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1Are you asking about this limit :https://math.stackexchange.com/questions/201906/showing-that-frac-sqrtnnn-rightarrow-frac1e? – StubbornAtom May 13 '18 at 14:27
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Stirling approximation gives a nice solution. – szw1710 May 13 '18 at 21:00