<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%% LyX 1.1 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[11pt,oneside]{amsart}
\usepackage[T1]{fontenc}
\usepackage{geometry}
\geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\setlength\parskip{\medskipamount}
\setlength\parindent{0pt}
\usepackage{graphics}

\makeatletter


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
\providecommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
 \theoremstyle{plain}    
 \newtheorem{thm}{Theorem}[section]
 \numberwithin{equation}{section} %% Comment out for sequentially-numbered
 \numberwithin{figure}{section} %% Comment out for sequentially-numbered
 \theoremstyle{definition}
  \newtheorem{example}[thm]{Example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\setcounter{section}{1}
\setcounter{subsection}{0}
\secounter{example}{2}
\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle
\begin{example}
Let \[
f(x)=\left\{ \begin{array}{cc}
x^{2} &amp; \mathrm{if}\, x\neq 1\\
5 &amp; \mathrm{if}\, x=1
\end{array}\right. .\]
Then \[
\lim _{x\rightarrow 1}f(x)=1,\]
 even though \( f(1)=5 \). 
\end{example}

\subsubsection*{Solution}

Here the function is purposely defined to have the \textit{wrong} value at 1.
The limit \[
\lim _{x\rightarrow 1}f(x)\]
 is what \( f(1) \)  \textit{ought to} be.  

\vspace{0.3cm}
{\par\centering \includegraphics{limits-basic-ex3.ps} \par}
\vspace{0.3cm}

This example stresses the predictability of values of the function. This function
is bad in that it doesn't have the expected value at 1. The previous example
had no value that \( f(1) \) should be; from one side it should be one thing,
from the other side it should be another. But \( f \) has to have just one
value. 


\subsubsection*{Back}
\end{document}
</pre></body></html>