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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{Infinite Limits}

Infinity (\( \infty  \)) occurs many times in calculus. It's not just some
sort of science-fiction invention, it has meaning, \emph{in terms of limits}.
In the case of  \[
\lim _{x\rightarrow \infty }f(x)=L\]


it means that, as \( x \) gets larger without bound, \( f(x) \) settles down
and gets closer to \( L \).  

\vspace{0.3cm}
{\par\centering \resizebox*{3in}{1.5in}{\includegraphics{horiz-asymptote.ps}} \par}
\vspace{0.3cm}

Similarly,  \[
\lim _{x\rightarrow \infty }f(x)=\infty \]


means that, as \( x \) gets larger and larger, so does \( f(x) \).  

Finally, it might happen that a function might itself get large without bound,
as \( x \) goes to a real number \( a \). We then say that  \[
\lim _{x\rightarrow a}f(x)=\infty .\]


As you might guess, the stuff about one-sided limits {[}Link{]} applies here
as well. Also, you can be more specific, in some cases, and talk about limts
as \( x \) goes to \( +\infty  \) or \( -\infty  \), or limits being \( +\infty  \)
or \( -\infty  \). This has specifirc implications on the general {}``shape{}''
of the function, and can help draw the graph \( y=f(x) \). 

None of this explanation really offers much insight on how you can compute limits
of more than the simplist examples. There are a few standard theorems and techniques
that make these computations straightforward. 

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

\end{example}

\subsubsection*{Solution}

\begin{eqnarray*}
\lim _{x\rightarrow \infty }\frac{\sqrt{x^{2}+1}-1}{x} & = & \lim _{x\rightarrow \infty }\frac{\left( \sqrt{x^{2}+1}-1\right) \frac{1}{x}}{\left( x\right) \frac{1}{x}}\, (\textrm{multiply}\, \textrm{top}\, \textrm{and}\, \textrm{bottom}\, \textrm{by}\, x)\\
 & = & \lim _{x\rightarrow \infty }\frac{\left( \sqrt{\frac{x^{2}+1}{x^{2}}}-\frac{1}{x}\right) }{1}\, ((\textrm{distribute})\\
 & = & \lim _{x\rightarrow \infty }\sqrt{\frac{x^{2}+1}{x^{2}}}-\frac{1}{x}\, (\textrm{The}\, \textrm{denominator}\, \textrm{is}\, 1\, \textrm{now},\, \textrm{so}\, \textrm{drop}\, \textrm{it})\\
 & = & \lim _{x\rightarrow \infty }\sqrt{1+\frac{1}{x^{2}}}-\frac{1}{x}\, (\textrm{simplify}\, \textrm{in}\, \textrm{the}\, \textrm{radical})\\
 & = & \sqrt{1+0}-0\, (\textrm{Finally}\, \textrm{take}\, \textrm{the}\, \textrm{limit}).\\
 & = & 1
\end{eqnarray*}



\subsubsection*{Back}
\end{document}
