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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Calculating Limits Using the Limit Laws">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 18:37:05 1997}
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\begin{document}


\section{Calculating Limits Using the Limit Laws\label{Limit laws}}

\subsubsection{Limit Laws}

\begin{theorem}
Suppose that $c$ is a constant and the limits 
\[
\lim_{x\stackrel{}{\rightarrow }a}f(x)\qquad \text{and\qquad }\lim_{x%
\stackrel{}{\rightarrow }a}g(x)
\]
\end{theorem}

exist. Then

\begin{enumerate}
\item[1.]  $\lim_{x\stackrel{}{\rightarrow }a}\lbrack f(x)+g(x)\rbrack
=\lim_{x\stackrel{}{\rightarrow }a}f(x)+\lim_{x\stackrel{}{\rightarrow }%
a}g(x)$

\item[2.]  $\lim_{x\stackrel{}{\rightarrow }a}\lbrack f(x)-g(x)\rbrack
=\lim_{x\stackrel{}{\rightarrow }a}f(x)-\lim_{x\stackrel{}{\rightarrow }%
a}g(x)$

\item[3.]  $\lim_{x\stackrel{}{\rightarrow }a}\lbrack cf(x)\rbrack =c$ $%
\lim_{x\stackrel{}{\rightarrow }a}f(x)$

\item[4.]  $\lim_{x\stackrel{}{\rightarrow }a}\lbrack f(x)g(x)\rbrack
=\lim_{x\stackrel{}{\rightarrow }a}f(x)\cdot \lim_{x\stackrel{}{\rightarrow }%
a}g(x)$

\item[5.]  $\lim_{x\stackrel{}{\rightarrow }a}\dfrac{f(x)}{g(x)}=\dfrac{%
\lim_{x\stackrel{}{\rightarrow }a}f(x)}{\lim_{x\stackrel{}{\rightarrow }%
a}g(x)}$\quad if $\lim_{x\stackrel{}{\rightarrow }a}g(x)\neq 0$

\item[6.]  $\lim_{x\stackrel{}{\rightarrow }a}\left[ f(x)\right] ^{n}=\left[
\lim_{x\stackrel{}{\rightarrow }a}f(x)\right] ^{n}$\qquad where $n$ is a
positive integer

\item[7.]  $\lim_{x\stackrel{}{\rightarrow }a}c=c$

\item[8.]  $\lim_{x\stackrel{}{\rightarrow }a}x=a$

\item[9.]  $\lim_{x\stackrel{}{\rightarrow }a}x^{n}=a^{n}$ where $n$ is a
positive integer

\item[10.]  $\lim_{x\stackrel{}{\rightarrow }a}\sqrt[n\,]{x}=\sqrt[n\,]{a}%
\qquad $where $n$ is a positive integer (If $n$ is even, we assume that $a>0$%
.)

\item[11.]  $\lim_{x\stackrel{}{\rightarrow }a}\sqrt[n\,]{f(x)}=\sqrt[n\,]{%
\lim_{x\stackrel{}{\rightarrow }a}f(x)}\qquad $where $n$ is a positive
integer (If $n$ is even, we assume that $\lim_{x\stackrel{}{\rightarrow }%
a}f(x)>0.$)

\item  If $f$ is a polynomial or a rational function and $a$ is in the
domain of $f$, then
\end{enumerate}

\[
\lim\limits_{x\stackrel{}{\rightarrow }a}f(x)=f(a) 
\]

\subsubsection{Limit Inequality}

\begin{theorem}
If $f(x)\leq g(x)$ for all $x$ in an open interval that contains $a$ (except
possibly at $a$) and the limits of $f$ and $g$ both exist as $\,x$
approaches $a$, then 
\[
\lim_{x\stackrel{}{\rightarrow }a}f(x)\leq \lim_{x\stackrel{}{\rightarrow }%
a}g(x) 
\]
\end{theorem}

\subsubsection{Squeeze Theorem\label{Squeeze theorem}}

\begin{theorem}
If $f(x)\leq g(x)\leq h(x)$ for all $x$ in an open interval that contains $a$
(except possibly at $a$) and 
\[
\lim_{x\stackrel{}{\rightarrow }a}f(x)=\lim_{x\stackrel{}{\rightarrow }%
a}h(x)=L 
\]
\end{theorem}

then 
\[
\lim_{x\stackrel{}{\rightarrow }a}g(x)=L 
\]

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

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

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

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

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

\noindent \hrulefill

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\noindent \hrulefill

\noindent

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