TITLE: Limit of n-th root of $1/n$
QUESTION [8 upvotes]: I'm struggling to calculate the limit
$$\lim_{n\rightarrow\infty}\left(\frac{1}{n}\right)^{\frac{1}{n}}$$
I have tried taking log,
$\log(\frac{1}{n})^{\frac{1}{n}}=\frac{1}{n}\log({\frac{1}{n}})$
and setting $t=\frac{1}{n}$ and rewite the desired limit as
$e^{\lim_{t\rightarrow 0}t\log(t)}$,
and I'm stuck here because $\log$ is not defined at $0$.
Did I miss something?
Thank you for reading.

REPLY [3 votes]: Just for the fun of it, here is another one:
Consider $\sum\limits_{n=1}^{\infty}\frac{1}{n}x^{n}$. This diverge at $x=1$ and converge at $x=-1$, hence the radius of convergence must be $1$. Using root test on this series give the radius of convergence to be $\frac{1}{\limsup\limits_{n\rightarrow\infty}(\frac{1}{n})^{\frac{1}{n}}}$ which show that $\limsup\limits_{n\rightarrow\infty}(\frac{1}{n})^{\frac{1}{n}}=1$.
Now consider $x^{x}=e^{x\ln(x)}$. Differentiate give $(1+\ln(x))e^{x\ln(x)}$ which is negative as long as $\ln(x)<-1$ which is $x<\frac{1}{e}$. Hence $(\frac{1}{n})^{\frac{1}{n}}$ is eventually monotone. That combine with $\limsup\limits_{n\rightarrow\infty}(\frac{1}{n})^{\frac{1}{n}}=1$ showed that $\lim\limits_{n\rightarrow\infty}(\frac{1}{n})^{\frac{1}{n}}=1$.