Saturday, 21 February 2015

real analysis - limit of nth root of factorial devided by n








I have a litle discution with a friend about the folowing limit:
lim
I would solve it like this: \lim_{n\to\infty} \sqrt[n]{\frac{n!}{n^{n}}} =0
or
\lim_{n\to\infty} \frac{\sqrt[n]{n}\sqrt[n]{n-1}\cdots\sqrt[n]{1}}{n}=\frac{1*1*1\cdots}{\infty}=0
and in this 2º way would there be a problem with 1^{\infty}? I would say that no, because there is no functions involved, since as much as I know this undetermination is because you would whant to avoid the situation such as f(x)^{g(x)} where f(x)\to1 and g(x)\to\infty Could anyone clarify this for me?

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