<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\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)]
$\log\left(\ds\frac{1}{\varepsilon}\right)&gt;\log\left(\log\ds\frac{1}{\varepsilon}\right)
&gt; 1
&gt;\varepsilon^{\frac{1}{2}}\log\left(\ds\frac{1}{\varepsilon}\right)&gt;\varepsilon^{\frac{1}{2}}
&gt;\varepsilon\log\left(\ds\frac{1}{\varepsilon}\right) &gt;
\varepsilon &gt; \varepsilon^{\frac{3}{2}} &gt;
\varepsilon^2\log\left(\ds\frac{1}{\varepsilon}\right) &gt;
\varepsilon^2 &gt;e^{-\frac{1}{\varepsilon}}$

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


\end{document}
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