%% 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{xca}[section]{Exercise}%%Delete [section] for sequential numbering

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\setcounter{section}{1}
\setcounter{subsection}{2}

\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\subsection{Infinite Limits}

\begin{xca}
\[
\lim _{x\rightarrow \infty }\sqrt{x^{2}+2x-1}-x=\]

\end{xca}

\subsubsection*{Hint}

Here the trick is what is called conjugation. Since \[
(a-b)(a+b)=a^{2}-b^{2},\]
whenever we have a difference of square roots, \[
(\sqrt{a}-\sqrt{b}),\]
we can multiply by the sum of those square roots, top and bottom, and rationalize
the numerator,\footnote{%
Most of you were taught to always rationalize the denominator of fractions.
I never understoond why you should do that. In many, many cases it makes much
more sense to rationalize the \emph{numerator}, since it is numerators that
are added together. 
} \begin{eqnarray*}
\left( \sqrt{a}-\sqrt{b}\right)  & = & \frac{\left( \sqrt{a}-\sqrt{b}\right) \left( \sqrt{a}+\sqrt{b}\right) }{\left( \sqrt{a}+\sqrt{b}\right) }\\
 & = & \frac{\left( a-b\right) }{\left( \sqrt{a}+\sqrt{b}\right) }.
\end{eqnarray*}
Applied to this case, \begin{eqnarray*}
\sqrt{x^{2}+2x-1}-x & = & \frac{\left( \sqrt{x^{2}+2x-1}-x\right) \left( \sqrt{x^{2}+2x-1}-x\right) }{\sqrt{x^{2}+2x-1}-x}\\
 & = & \frac{\left( (x^{2}+2x-1)-x^{2}\right) }{\sqrt{x^{2}+2x-1}-x}.
\end{eqnarray*}



\subsubsection*{Back}
\end{document}
