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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Limits at Infinity; Horizontal Asymptotes">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 18:38:28 1997}
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\begin{document}


\section{\label{Limits at infinity}\label{Horizontal asymptotes}Limits at
Infinity; Horizontal Asymptotes}

\subsubsection{Horizontal Asymptote}

\begin{definition}
The line $y=L$ is called a \textbf{horizontal asymptote} of the curve $y=f(x)
$ if either 
\[
\lim\limits_{x\rightarrow \infty }\,\,f(x)=L\qquad \text{or}\qquad
\lim\limits_{x\rightarrow \,-\infty }\,\,f(x)=L
\]
\end{definition}

\begin{example}
The line $y=1$ is a horizontal asymptote of the curve $y=\frac{x^{2}+3x-1}{%
x^{2}-2},$ as depicted in the following graph.\FRAME{dtbpFU}{3in}{2.0003in}{%
0pt}{\Qcb{$\lim_{x\stackrel{}{\rightarrow }\infty }\frac{x^{2}+3x-1}{x^{2}-2}%
=1$}}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width
3in;height 2.0003in;depth 0pt;display "PICT";function
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50;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}
\end{example}

\subsubsection{Limits at Infinity}

\begin{theorem}
If $r>0$ is a rational number, then 
\[
\lim\limits_{x\rightarrow \infty }\,\,\frac{1}{x^{r}}=0 
\]
If $r<0$ is a rational number such that $x^{r}$ is defined for all $x$, then 
\[
\lim\limits_{x\rightarrow \,-\,\infty }\,\,\frac{1}{x^{r}}=0 
\]
\end{theorem}

\subsection{\ Related Reference Topics}

{\small \hyperref{Algebra}{}{}{Al_contents.tex}}

{\small \hyperref{Geometry}{}{}{GE_contents.tex}}

{\small \hyperref{Trigonometry}{}{}{TR_contents.tex}}\hfill

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Definition and notation for limits}{}{}{%
../../Help/DM7-2.tex#Notation for limits}}

{\small \hyperref{Infinite limits }{}{}{../../Help/DM7-2.tex#Infinite limits}%
}

{\small \hyperref{Limits}{}{}{../../Help/DM7-2.tex#Limit}}

{\small \hyperref{Limits at infinity}{}{}{../../Help/DM7-2.tex#Limits at
infinity}}

{\small \hyperref{One-sided limits}{}{}{../../Help/DM7-2.tex#One-sided limits%
}}

{\small \hyperref{Plotting Curves and Surfaces}{}{}{../../Help/DM6.tex}}

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