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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\subsubsection{Examples of somputing limits using rules and theorems.}

\begin{example}
Compute 
\[
\lim _{x\rightarrow 3}\frac{x^{2}+2x-1}{x^{3}-3x^{2}+2}.\]

\end{example}
%

\begin{example}
Compute 
\[
\lim _{x\rightarrow -1}\sqrt{x^{3}+2x+7}.\]

\end{example}
\begin{proof}
To be fair, we really don't have the theory to claim this (yet), but we can
argue simply that, since 
\begin{eqnarray*}
\lim _{x\rightarrow -1}x^{3}+2x+7 & = & \lim _{x\rightarrow -1}\left( \sqrt{x^{3}+2x+7}\right) ^{2}\\
 & = & \lim _{x\rightarrow -1}\sqrt{x^{3}+2x+7}\lim _{x\rightarrow -1}\sqrt{x^{3}+2x+7},
\end{eqnarray*}
and 
\[
\lim _{x\rightarrow -1}x^{3}+2x+7=(-1)^{3}+2(-1)+7=4,\]
 then 
\[
\lim _{x\rightarrow -1}\sqrt{x^{3}+2x+7}=2.\]

\end{proof}
\begin{example}
Compute 
\[
\lim _{x\rightarrow 9}\frac{x^{2}-81}{\sqrt{x}-3}.\]

\end{example}
%

\begin{example}
Define 
\[
f(x):=\left\{ \begin{array}{ll}
\sqrt{x-4}, & \mathrm{if}\, x>4\\
8-2x, & \mathrm{if}\, x<4
\end{array}\right. .\]
Find 
\[
\lim _{x\rightarrow 4}f(x),\]
 if it exists.
\end{example}
%

\begin{example}
Find 
\[
\lim _{x\rightarrow 0}x\sin (\frac{1}{x})\]
 by using the squeeze theorem.

\end{example}
\end{document}
