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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\subsection{Limits with infinity}

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

\end{example}
%

\begin{example}
\[
\lim _{x\rightarrow \infty }\frac{x^{3}+x^{2}x-6}{x^{3}+5}\]

\end{example}
%

\begin{example}
\[
\lim _{x\rightarrow 1-}\frac{1}{x-1}\]

\end{example}
%

\begin{example}
\[
\lim _{x\rightarrow \infty }\frac{x^{2}+5x-6}{x^{3}+x+5}\]

\end{example}

\subsubsection{Examples of limits by the definition}

\begin{example}
Show that 
\[
\lim _{x\rightarrow 1}3x+5=8.\]
 
\end{example}
%

\begin{example}
Show that 
\[
\lim _{x\rightarrow 2}x^{2}+3x=10.\]
 
\end{example}
%

\begin{example}
Show that 
\[
\lim _{x\rightarrow 3}\frac{1}{x}=\frac{1}{3}.\]
 
\end{example}
%

\begin{example}
Find a number \( \delta  \) sufficiently small so that the distance from 
\[
f(x)=2x^{2}+3x-1\]
 to \( 4 \) is less than \( 1/100 \) if \( |x-1|<\delta  \). 

\end{example}
\end{document}
