Question

Calculate
\[\lim_{n\to\infty}(\sqrt[n]{n}-1)^n\]

Answer 1

\[\lim_{n\to\infty}(\sqrt[n]{n}-1)^n\]
\[\lim_{n\to\infty}(n ^{\frac{1}{n}}-1)^n\]
Try a the smaller limit of n^(1/n)
\[\lim_{n\to\infty}n ^{\frac{1}{n}} =\infty^{0}\]
Anything to the 0th power is 1, so this piece of the limit is slowly approaching 1.
Therefore, you are left with:
\[\lim_{n\to\infty}(1-1)^n = \lim_{n\to\infty}0^n = 0\]

The answer is 0. I don't think I did a very good job of using limits though.
http://www.wolframalpha.com/input/?i=Plot%5B%28n^%281%2Fn%29+-+1%29^n%2C+ {n%2C0%2C100}%5D