\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\newcommand{\ud}{\underline}
\newcommand{\undb}{\underbrace}
\parindent=0pt
\begin{document}


{\bf Question}

Arrange the following in descending order for small positive
$\varepsilon$:

\begin{description}
\item[(a)]
$\varepsilon^2,\ \varepsilon^{\frac{1}{2}},\
\log\left(\log\ds\frac{1}{\varepsilon}\right),\ 1,\
\varepsilon^{\frac{1}{2}}\log\left(\ds\frac{1}{\varepsilon}\right),$

$\varepsilon\log\left(\ds\frac{1}{\varepsilon}\right),\
e^{-\frac{1}{\varepsilon}},
\log\left(\ds\frac{1}{\varepsilon}\right),\
\varepsilon^{\frac{3}{2}},\ \varepsilon,\
\varepsilon^2\log\left(\ds\frac{1}{\varepsilon}\right).$

\item[(b)]
$e^{-\frac{1}{\varepsilon}},\
\log\left(\ds\frac{1}{\varepsilon}\right),\ \varepsilon^{-0.01},\
\cot\varepsilon,\ \sinh\left(\ds\frac{1}{\varepsilon}\right).$
\end{description}

\vspace{.5in}

{\bf Answer}

Try them on your calculator for $\varepsilon \to 0+$ and then use
limits to justify them

\begin{description}
\item[(a)]
$\log\left(\ds\frac{1}{\varepsilon}\right)>\log\left(\log\ds\frac{1}{\varepsilon}\right)
> 1
>\varepsilon^{\frac{1}{2}}\log\left(\ds\frac{1}{\varepsilon}\right)>\varepsilon^{\frac{1}{2}}
>\varepsilon\log\left(\ds\frac{1}{\varepsilon}\right) >
\varepsilon > \varepsilon^{\frac{3}{2}} >
\varepsilon^2\log\left(\ds\frac{1}{\varepsilon}\right) >
\varepsilon^2 >e^{-\frac{1}{\varepsilon}}$

\item[(b)]
$\sinh\left(\ds\frac{1}{\varepsilon}\right)>\cot\varepsilon>
\varepsilon^{-0.01}>e^{-\frac{1}{\varepsilon}}>
\log\left(\ds\frac{1}{\varepsilon}\right)>e^{-\frac{1}{\varepsilon}}$
\end{description}


\end{document}
