Show that $\left\{ \dfrac{n}{e^{n}}\right\}$ converges and find its limit.

Solution We begin with the related function $f(x)=\dfrac{x}{e^{x}},$ $x>0$. To find $\lim\limits_{x\rightarrow \infty }f(x)$, we use L’Hôpital’s Rule. $$\begin{eqnarray*} \lim_{x\,\rightarrow \,\infty }\,f(x) &=& \lim_{x\,\rightarrow \,\infty }\frac{x}{e^{x}} \underset{\underset{\color{#0066A7}{\rm Use L’Hôpital’s Rule}}{\color{#0066A7}{\uparrow}}}{=} \lim_{x\,\rightarrow\,\infty }\frac{1}{e^{x}}=0 \\[-8pt] \end{eqnarray*}$$

Since $\lim\limits_{x\,\rightarrow \,\infty }\,f(x)=0$, the sequence $\left\{ \dfrac{n}{e^{n}}\right\}$ converges and $\lim\limits_{n\,\rightarrow \,\infty}\dfrac{n}{e^{n}}=0$.