Show that this limit is equal to $\liminf  a_{n}^{1/n}$ for positive terms. 
Show that if $a_{n}$ is a sequence of positive terms such that $\lim\limits_{n\to\infty} (a_{n+1}/a_n) $ exists, then this limit is equal to $\liminf\limits_{n\to\infty} a_n^{1/n}$.

I am not event sure where to start from, any help would be much appreciated.

Hint:
$$a_n ^{\frac{1}{n}}= e^{\frac{ \ln (a_n)}{n}}$$
$$(a_{n+1}/a_n)= e^{\ln (a_{n+1}) -\ln(a_n)} \,.$$
Try to prove instead the equality of the corresponding exponents limits..
P.S. Are you familiar with Stolz Cezaro or Cezaro means?
