<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{\un}{\underline}
\parindent=0pt
\begin{document}

{\bf Question}

Using the $\delta-\epsilon$ definition of limit, prove that for
each $a\in {\bf C} -\{ 0\}$,
\[ \lim_{z\rightarrow\infty} az =\infty. \]
\medskip

{\bf Answer}

$\ds\lim_{z \to \infty} az=\infty$ ($a \ne 0$)

given $\epsilon&gt;0$, there is $\delta&gt;0$ so that if $z \in
\cup_{\delta} (\infty)-\{\infty\}$ then $az \in
\cup_{\epsilon}(\infty)$.

$$az \in \cup_{\epsilon}(\infty):\ |az|&gt;\epsilon\ \
|z|&gt;\ds\frac{\epsilon}{|a|}$$

Take $\delta=\ds\frac{\epsilon}{|a|}$: then if $z \in
\cup_{\delta} (\infty)-\{\infty\}$ then $|z|&gt;\delta$ and so
$|z|&gt;\ds\frac{\epsilon}{|a|}$ and so $|az|&gt;\epsilon$ as desired.
\end{document}
</pre></body></html>