%% 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}

\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}{4}
\setcounter{subsection}{3}

\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\subsection{l'H\^{o}pital's Rule}

\begin{example}
\[
\lim _{x\rightarrow 0}\frac{\sin (3x)}{\tan (4x)}\]
 
\end{example}

\subsubsection*{Solution}

This limit can be found by more elementary means, but it certainly is easy using
l'H\"{o}pital's rule:

\footnote{%
Many textbooks insist on using a special notation for the equals sign when the
step involves l'H\"{o}pital's rule, so that this would look like:\[
\lim _{x\rightarrow 0}\frac{\sin (3x)}{\tan (4x)}\begin{array}{c}
\begin{array}{c}
*\\
=\\

\end{array}
\end{array}\lim _{x\rightarrow 0}\frac{3\cos (3x)}{4\sec ^{2}(4x)},\]
or some similar notation. However, no one seems to think it is necessary for
a special equals sign to indicate any other particular step has been taken,
so it seems odd to do so here. Truth be known, many mathematicians think that
l'H\"{o}pital's rule is abused by students, since it is used when more straightforward
calculation might be more appropriate. It is this prejudice which impels them
to use a special indicator when applying this theorem, as if it were a trick
not entirely to be trusted.
}\begin{eqnarray*}
\lim _{x\rightarrow 0}\frac{\sin (3x)}{\tan (4x)} & = & \lim _{x\rightarrow 0}\frac{3\cos (3x)}{4\sec ^{2}(4x)}\\
 & = & \frac{3}{4},
\end{eqnarray*}
since \[
\lim _{x\rightarrow 0}\cos (3x)=1=\lim _{x\rightarrow 0}\sec ^{2}(4x).\]



\end{document}
