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\makeatletter


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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  \[
\lim _{x\rightarrow a}f(x)=\infty ,\]
 or that \( f \) has a vertical asymptote at \( x=a \), which might look like
one of these graphs

\vspace{0.3cm}
{\par\centering \includegraphics{vert-asymptotes.ps} \par}
\vspace{0.3cm}

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*{Answer}


\subsubsection*{Solution}

\begin{example}
\[
\lim _{x\rightarrow \infty }\frac{2x^{3}+x^{2}+x-6}{x^{3}+4x+5}\]

\end{example}

\subsubsection*{Hints}

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

\end{example}

\subsubsection*{Hints}

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

\end{example}

\subsubsection*{Hints}

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

\end{xca}

\subsubsection*{Hint}

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

\end{xca}
%

\begin{xca}
\[
\lim _{x\rightarrow 1+}\frac{x^{2}-5x+6}{x-1}=\]

\end{xca}
%

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

\end{document}
