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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{One-sided limits }

For the function \[
f(x)=\left\{ \begin{array}{ll}
3x+1, & \mathrm{if}\, x\geq 1\\
x+2, & \mathrm{if}\, x<1
\end{array}\right. ,\]
 

\vspace{0.3cm}
{\par\centering \resizebox*{2in}{2in}{\includegraphics{one-sided-limit.ps}} \par}
\vspace{0.3cm}

as far as the previous definition goes, there is no limit. As you approach \( x=1 \)
from the left (\( x \) near \( 1 \) but \( x<1 \)), the value of \( f(x) \)
is approaching \( 3 \) (following the red part of the graph), but if you approach
from the right (\( x \) near \( 1 \) but \( x>1 \)), the value of \( f(x) \)
is approaching \( 4 \), following the blue part of the graph. There really
are limits at 1, as long as you don't approach 1 from both sides. That's called
a \textit{one-sided} limit:  

\begin{defn}
We say that the limit from the left (or below) of  \( f(x) \) at \( a \) exists
and equals \( L \), and write  \[
\lim _{x\rightarrow a-}f(x)=L\, \, \left( \mathrm{or}\, \lim _{x\uparrow a}f(x)=L\right) ,\]
 if we can make \( f(x) \) as close as we need to \( L \) just by taking \( x \)
close enough to \( a \), keeping \( x<a \). 
\end{defn}
You can imagine how we define  \[
\lim _{x\rightarrow a+}f(x)=L\, \, \left( \mathrm{or}\, \lim _{x\downarrow a}f(x)=L\right) .\]


In order to have a real limit at \( a \), it is enough to have both one sided
limits as \( x \) approaches \( a \) from above and from below, and the two
one-sided limits have to be the same. That is: 

\begin{thm}
\[
\lim _{x\rightarrow a}f(x)=L\, \Leftrightarrow \, \lim _{x\rightarrow a+}f(x)=L\, \, \mathrm{and}\, \lim _{x\rightarrow a-}f(x)=L.\]

\end{thm}
\begin{example}
Find \[
\lim _{x\uparrow 1}\frac{x^{2}-1}{\left| x-1\right| }.\]

\end{example}

\subsubsection*{Answer }


\subsubsection*{Solution}

\begin{xca}
If \[
f(x)=\left\{ \begin{array}{ll}
3x^{2}+x, & \mathrm{if}\, x\geq 1\\
4x, & \mathrm{if}\, x<1
\end{array}\right. ,\]
find all of the following that exist:\[
\lim _{x\uparrow 1}f(x)=\]
\[
\lim _{x\downarrow 1}f(x)=\]
\[
\lim _{x\rightarrow 1}f(x)=\]

\end{xca}
%

\begin{xca}
If \[
f(x)=\left\{ \begin{array}{ll}
3x^{2}+x, & \mathrm{if}\, x>2\\
4x, & \mathrm{if}\, x\leq 2
\end{array}\right. ,\]
find all of the following that exist:\[
\lim _{x\uparrow 1}f(x)=\]
\[
\lim _{x\downarrow 1}f(x)=\]
\[
\lim _{x\rightarrow 1}f(x)=\]
\end{xca}

\end{document}
