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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\section{Limits and continuity}


\subsection{The idea of a limit\label{limit-idea}}

This section explains the basic idea of a limit, which you need to understand
in order for the rest of this topic to make sense. Briefly, the limit of a function
\( f(x) \) as \( x \) appoaches \( a \), 
\[
\lim _{x\rightarrow a}f(x),\]
is the number that \( f(a) \) \emph{should} be.


\subsection{One-sided limits \label{limit-one-sided}}

Some functions don't have a real limitas \( x \) appoaches \( a \), but it
may still be possible to makes sense of what happens to the function \( f(x) \)
as you approach \( a \) from only one side.


\subsection{Infinite Limits\label{limits-infinite}}

The notion of ``infinity'' has been much-abused by philosophers and science-fiction
writers. But, really, all that inifinity (\( \infty  \)) means mathematically
is a certain kind of limit. Two distinct kinds of limits, actually.


\subsection{\label{limits-computing}Computing limits \protect\url{limits-computing.html}}

Once you have some grounding in the idea of what limits are supposed to be,
this section gives a number of methods to actually compute limits in a number
of cases. Some of these methods will come back to haunt us later on, since we
really do need to compute limits to make sense of some of the later formulas
we develop. This also includes some basic theorems about limits. 


\subsection{Theory of limits\label{limits-theory}}

The definition of a limit used in the first section (\ref{limit-idea}) of this
topic was correct, but used imprecise terms that you probably haven't seen in
a mathematics course before. The idea of a limit was around a lot longer than
any formal definition, but this definintion does put it on a firm mathematical
base. This section deals with that formal definition, and you work through some
examples of how to apply the definition. 


\subsection{Continuity\label{continuity}}

Simply put, a function \( f(x) \) is \emph{continuous} (\emph{at} a point \( x=a \))
if \( f(a) \) has the value that it should have, 
\[
\lim _{x\rightarrow a}f(x)=f(a).\]
We think of continuous functions (at all points) as ``good'' functions, since
those are the ones we can deal with in calculus. This section introduces the
concept.


\subsection{Theorems about continuous functions\label{continuity-theorems}}

These basic theorems seem to be obvious, but aren't, quite, as obvious as they
seem, since the only need to be true if the function is continuous. They provide
the theoretical basis for much of the theory of calculus.

\end{document}
