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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/The Precise Defintion of a Limit">}
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\begin{document}


\section{The Precise Definition of a Limit\label{Limit}}

\subsubsection{Limit}

\begin{definition}
Let $f$ be a function defined on some open interval that contains the number 
$a$, except possibly at $a$ itself. Then we say that the\textbf{\ limit of }$%
f(x)$\textbf{\ as }$x$\textbf{\ approaches }$a$\textbf{\ is }$L$, and we
write 
\[
\lim_{x\stackrel{}{\rightarrow }a}f(x)=L
\]
if for every number $\varepsilon >0$ there is a corresponding number $\delta
>0$ such that 
\[
\left| f(x)-L\right| <\varepsilon \qquad \text{whenever\qquad }0<\left|
x-a\right| <\delta 
\]
\end{definition}

\subsubsection{\label{Limit, left-hand}Limit, Left-Hand}

\begin{definition}
Let $f$ be a function defined on some half-open interval with $a$ as right
endpoint. Then we say that the\textbf{\ limit of }$f(x)$\textbf{\ as }$x$%
\textbf{\ approaches }$a$\textbf{\ from the left is }$L$, and we write 
\[
\lim_{x\stackrel{}{\rightarrow }a^{-}}\,f(x)=L
\]
if for every number $\varepsilon >0$ there is a corresponding number $\delta
>0$ such that 
\[
\;\left| f(x)-L\right| <\varepsilon \qquad \text{whenever\qquad }a-\delta
<x<a
\]
\end{definition}

\subsubsection{\label{Limit, right-hand}Limit, Right-Hand}

\begin{definition}
Let $f$ be a function defined on some half-open interval with $a$ as left
endpoint. Then we say that the\textbf{\ limit of }$f(x)$\textbf{\ as }$x$%
\textbf{\ approaches }$a$\textbf{\ from the right is }$L$, and we write
\end{definition}

\[
\lim_{x\rightarrow a^{+}}\,f(x)=L 
\]
if for every number $\varepsilon >0$ there is a corresponding number $\delta
>0$ such that 
\[
\left| \,f(x)-L\,\right| <\varepsilon \qquad \text{whenever\qquad }%
a<x<a+\delta 
\]
\bigskip

The following test for limits is sometimes useful.

\begin{theorem}
$\lim_{x\stackrel{}{\rightarrow }a}\,f(x)=L$ if and only if $\lim_{x%
\stackrel{}{\rightarrow }a^{-}}\,f(x)=L$ and $\lim_{x\stackrel{}{\rightarrow 
}a^{+}}\,f(x)=L$.\bigskip
\end{theorem}

\begin{example}
The limit $\lim_{x\rightarrow 0}\dfrac{x}{\left| x\right| }$ does not exist
because $\lim_{x\rightarrow 0^{-}}\dfrac{x}{\left| x\right| }=-1$ while $%
\lim_{x\rightarrow 0^{+}}\dfrac{x}{\left| x\right| }=1$\FRAME{dtbpFU}{3in}{%
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\end{example}

\subsubsection{\label{Limit, infinite positive}Limit, Infinite Positive}

\begin{definition}
Let $f$ be a function defined on some open interval that contains the number 
$a$, except possibly at $a$ itself. Then 
\[
\lim_{x\stackrel{}{\rightarrow }a}f(x)=\infty 
\]
means that for every positive number $M$ there is a corresponding number $%
\delta >0$ such that 
\[
f(x)>M\qquad \text{whenever}\qquad 0<\left| x-a\right| <\delta 
\]
\end{definition}

\subsubsection{\label{Limit, infinite negative}Limit, Infinite Negative}

\begin{definition}
Let $f$ be a function defined on some open interval that contains the number 
$a$, except possibly at $a$ itself. Then 
\[
\lim_{x\stackrel{}{\rightarrow }a}f(x)=-\,\infty 
\]
means that for every negative number $N$ there is a corresponding number $%
\delta >0$ such that 
\[
f(x)<N\qquad \text{whenever}\qquad 0<\left| x-a\right| <\delta 
\]
\end{definition}

\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{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%
}}

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