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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 \[
\lim _{x\rightarrow 3}\frac{1}{x}=\frac{1}{3}.\]
 
\end{example}

\subsubsection*{Hints}

Again, you need to do a ballpark estimate. But, first you want to look at what
you are trying to control, and re-express it to make \( |x-3| \) appear. 


\subsubsection*{Next step}

\[
\left| \frac{1}{x}-\frac{1}{3}\right| =\frac{|3-x|}{|3x|}.\]



\subsubsection*{And then?}

\begin{eqnarray*}
\left| \frac{1}{x}-\frac{1}{3}\right|  & = & \frac{|3-x|}{|3x|}\\
 & = & \frac{1}{3|x|}|x-3|.
\end{eqnarray*}
We need to control the \( 1/3|x| \) term, since the other one we have complete
control over. 


\subsubsection*{Finish it.}
\end{document}
