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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Precise Definition of a Limit}

\begin{example}
Show that \[
\lim _{x\rightarrow \infty }\sqrt{x}=\infty .\]

\end{example}

\subsubsection*{Solution }

Let \( E\geq 0 \) be given. Now, let's find the number \( D \). But how, you
might ask, can we find \( D \) if we don't know what \( E \) is, really? The
idea is to find a \( D \) expressed  \textbf{in terms of \( E \)} that works,
so that, no matter what \( E \) is, \( D \) is adjusted automatically. You
figure that out by a sort of reverse-engineering: In order to make \( \sqrt{x}>E \),
try to see if you can get an inequality involving \( x \) from the one involving
\( f(x) \). Here the way to do it is to square both sided of the inequality.
Since everything in the inequality is positive, that won't change the inequality
(you have to worry about that sort of thing, sometimes). Then: \[
\sqrt{x}>E\iff x>E^{2},\]


so, if \( x>E^{2} \), we'll guarantee that \( \sqrt{x}>E \). So, \( E^{2} \)
will do as our \( D \), \( D=E^{2} \). 

Note that I said that that choice of \( D \) \textit{will do}. There isn't
any  \textbf{one} answer here, you just need to find a way to get a large enough
\( x \);  \textit{larger} will also do nicely. 


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