Sequence Limit: $\lim\limits_{n \rightarrow \infty}{n\,x^n}$

Consider the series $\sum n x^n$. The ratio test shows that this series converges when $|x|<1$. Hence in this case the sequence of terms must converge to zero.

Hint: if $|x| < r < 1$, show that $|(n+1) x^{n+1}| < r\, |n x^n|$ for sufficiently large $n$.

If $x=0$ we're done. Otherwise, we can show absolute convergence using l'Hôpital's, $$\begin{eqnarray*} \lim_{n\to\infty}|n x^n| &=& \lim_{n\to\infty} \frac{n}{|x|^{-n}} \\ &=& \lim_{n\to\infty} \frac{\frac{d}{dn} n}{\frac{d}{dn} |x|^{-n}} \\ &=& \lim_{n\to\infty} \frac{1}{-|x|^{-n}\log |x|} \\ &=& \lim_{n\to\infty} -\frac{|x|^n}{\log |x|} \\ &=& 0. \end{eqnarray*}$$