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%TCIDATA{<META NAME="Title" CONTENT="Limits">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 18:15:07 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
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\input{tcilatex}
\begin{document}


\section{Limits 
\index{Limit}\label{Limit}}

The concept of a limit is fundamental to the study of calculus. It is the
central idea of the subject and is what distinguishes calculus from earlier
mathematics. The notion, which encompasses subtle concepts such as
instantaneous velocity, can be fully understood only through experience and
experimentation. With \textsl{Scientific Notebook}, you have a variety of
tools for computing and experimenting with limits.

The following examples illustrate that functions may have a limit at points
where they are not defined directly.

\begin{example}
If you define $f(x)=%
\frac{x^{2}-1}{x-1}$ and attempt to evaluate $f(1)$, you get an error
message because you are attempting division by zero. If you graph the
function $f(x)=\frac{x^{2}-1}{x-1}$, it appears to be well behaved for $x$
near $1$. Limits can be used to study the behavior of $f(x)$ for $x$ near $1$%
.\medskip 
\end{example}

\begin{example}
If you drop a pencil on a table, it first falls halfway to the table. Then
it falls halfway from that point to the table. Then it falls halfway from
that point to the table, and so on ad infinitum. This description might lead
you to believe it never reaches the table, but experience tells you
otherwise. Limits can be used to explain phenomena such as this.\medskip 
\end{example}

\subsection{\ Notation for Limits\label{Notation for limits}}

We say that the \emph{limit} of $f$ as $x$ approaches $a$ is $L$ and we
write $\lim_{x\rightarrow a}f(x)=L$ if for each \label{Epsilon}number $%
\varepsilon >0$ there exists a \label{Delta}number $\delta >0$ such that $%
\left| f(x)-L\right| <\epsilon $ whenever $0<\left| x-a\right| <\delta $.

\begin{quote}
$\blacktriangleright $ To find a limit of the form $\lim_{x\rightarrow
a}f(x) $ using \textsl{Scientific Notebook}
\end{quote}

\begin{enumerate}
\item[1.]  Type \QTR{UserInput}{lim} while in mathematics mode, or click 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{function.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'FUNCTION.WMF';file-properties "XNPEU";}}and
choose $\lim $.

\item[2.]  Click \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{subscrip.wmf%
}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.3009in;height 0.3009in;depth
0.0701in;original-width 17.8125pt;original-height 16.9375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'Subscrip.wmf';file-properties "XNPEU";}} and enter $x\rightarrow a$ as a
subscript.

\item[3.]  Enter $f(x)$.

\item[4.]  Choose \textsf{Evaluate}, press \textsc{ctrl + e}, or click 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{evaluate.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Evaluate.wmf';file-properties "XNPEU";}}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\lim\limits_{x\rightarrow 1}\dfrac{x^{2}-1}{x-1}=\allowbreak 2$
\end{quotation}

This result is reasonable, since for $x\neq 1$, $\frac{x^2-1}{x-1}%
=\allowbreak x+1$, which is close to 2 when $x$ is close to 1.

Limits of rational functions are not always apparent. You cannot evaluate
the following expression at $x=-3/2,$ because the denominator is 0 at this
point. However the expression has a limit at $-3/2$.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\lim\limits_{x\rightarrow -3/2}\dfrac{4x^{4}+6x^{2}+19x+6x^{3}+15}{%
2x^{3}+5x^{2}+5x+3}=\allowbreak -\dfrac{25}{7}$
\end{quotation}

\ 

\textsf{Factor} in place and \textsf{Simplify} in place allow you to fill in
the steps leading to evaluation by direct substitution, because the second
step removes the singularity from the expression. For the first two lines,
copy the entire expression after an equals sign, and carry out in-place
operations. For the third line, substitute $x=-3/2$ into the expression.

\begin{quote}
$\blacktriangleright $ \textsf{Factor}, \textsf{Simplify}, ..., \textsf{%
Evaluate}
\end{quote}

\begin{quotation}
\begin{eqnarray*}
\lim_{x\rightarrow -3/2}\frac{4x^{4}+6x^{2}+19x+6x^{3}+15}{2x^{3}+5x^{2}+5x+3%
} &=&\lim_{x\rightarrow -3/2}\frac{\left( 2x+3\right) \left( x+1\right)
\left( 2x^{2}-2x+5\right) }{\left( 2x+3\right) \left( x^{2}+x+1\right) }%
\vspace{6pt} \\
&=&\lim_{x\rightarrow -3/2}\left( x+1\right) \frac{2x^{2}-2x+5}{x^{2}+x+1}%
\vspace{6pt} \\
&=&\left[ \left( x+1\right) \frac{2x^{2}-2x+5}{x^{2}+x+1}\right] _{x=-3/2}%
\vspace{6pt} \\
&=&-\frac{25}{7} \\
&&
\end{eqnarray*}
\end{quotation}

\strut

\label{Substitution}You can carry out the \hyperref{substitution}{}{}{%
DM3-4.tex#Substitution} in the third line of the preceding example as
follows.

\begin{quote}
$\blacktriangleright $ To substitute a value into the expression
\end{quote}

\begin{enumerate}
\item[1.]  Select the expression $\left( x+1\right) \frac{2x^{2}-2x+5}{%
x^{2}+x+1}$ with the mouse or \textsc{shift + right arrow}.

\item[2.]  Click \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{sqbrack.wmf%
}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 0.3009in;height 0.3009in;depth
0.0701in;original-width 20.3125pt;original-height 18.8125pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'Sqbrack.wmf';file-properties "XNPEU";}}.

\item[3.]  Click \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{subscrip.wmf%
}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 0.3009in;height 0.3009in;depth
0.0701in;original-width 17.8125pt;original-height 16.9375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'Subscrip.wmf';file-properties "XNPEU";}}and enter $x=-3/2$ as a subscript.

\item[4.]  Choose \textsf{Evaluate}.\medskip
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left[ \left( x+1\right) \frac{2x^{2}-2x+5}{x^{2}+x+1}\right]
_{x=-3/2}=\allowbreak -\frac{25}{7}$
\end{quotation}

\label{Replace}You can also carry out the replacement in the third line
using the editing features of \textsl{Scientific Notebook}.

\begin{quote}
$\blacktriangleright $ To do an automatic replacement of mathematics
\end{quote}

\begin{enumerate}
\item[1.]  Select the expression $\left( x+1\right) \frac{2x^{2}-2x+5}{%
x^{2}+x+1}$ with the mouse or \textsc{shift + right arrow}.

\item[2.]  From the \textsf{Edit} menu, choose \textsf{Replace}.

\item[3.]  Fill in the choices in the dialog box in mathematics mode.

\begin{itemize}
\item  \textsf{Search for}: $x$

\item  \textsf{Replace with}: $(-3/2)$
\end{itemize}
\end{enumerate}

The result is the expression 
\[
\left( \left( -3/2\right) +1\right) \frac{2\left( -3/2\right) ^{2}-2\left(
-3/2\right) +5}{\left( -3/2\right) ^{2}+\left( -3/2\right) +1} 
\]

\strut

\subsection{\ Tables of Values and Plots \label{Table of values}}

The following limit is of special interest. We examine the behavior of this
function near $0$.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\lim\limits_{x\rightarrow 0}\dfrac{\sin x}{x}=\allowbreak 1$
\end{quotation}

\begin{example}
To see numerical evidence that $\lim_{x\rightarrow 0}\frac{\sin x}{x}%
=\allowbreak 1$, you can evaluate the expression $\frac{\sin x}{x}$ for
several values of $x$ near $0$. First define a function $f$ to be equal to
this expression so that it can be evaluated easily:

Place the insertion point in the equation $f(x)=\frac{\sin x}{x}$ and, from
the \textsf{Define} submenu, choose \textsf{New Definition}.

Applying \textsf{Evaluate Numerically} gives function values $%
f(0.1)=\allowbreak .9983341665$, $f(0.01)=\allowbreak .9999833334$, and $%
f(-0.001)=\allowbreak .9999998333$. Note that the function values seem to
approach 1.\medskip 
\end{example}

You can generate a table of values by applying a function to a vector of
domain values and then concatenating matrices, or you can do it in one step
by defining appropriate auxiliary functions. Following are examples of each.

\begin{quote}
$\blacktriangleright $ To create a table of values for the function $y=f(x)$
by defining auxiliary functions
\end{quote}

\begin{enumerate}
\item  Define the function $f(x)$

\item  Define a function $g(n)$ that will provide a sample of values of the
independent variable.

\item  Define the function $h(i,j)=(2-j)g(i)+(j-1)f(g(i))$

\item  From the Matrices submenu, choose \textsf{Fill Matrix.}\FRAME{dtbpF}{%
3.5051in}{2.06in}{0pt}{}{}{fillmat2.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file
"F";width 3.5051in;height 2.06in;depth 0pt;original-width
249.9375pt;original-height 146.0625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'FillMat2.wmf';file-properties "XNPEU";}}

\item  Set \textsf{Columns} to 2, and and set \textsf{Rows} to match the
size of your sample.

\item  Under \textsf{Fill With}, choose \textsf{Defined By Function}.

\item  In the input box for function name, enter $h$.

\item  Choose \textsf{OK}.
\end{enumerate}

\strut 

\begin{example}
Make a table of values for the sine function.
\end{example}

\begin{quote}
$\blacktriangleright $To create a table of values for the function $y=\sin x$
\end{quote}

\begin{enumerate}
\item  With the insertion point in the equation $f(x)=\sin x$, choose 
\textsf{Define + New Definition}.

\item  With the insertion point in the equation $g\left( i\right) =i\ast
10^{-2}$, choose \textsf{Define + New Definition}.

\item  With the insertion point in the equation $%
h(i,j)=(2-j)g(i)+(j-1)f(g(i))$, choose \textsf{Define + New Definition}.

\item  Choose expanding brackets and leave the insertion point in the input
box.

\item  From the \textsf{Matrices} submenu, choose \textsf{Fill Matrix.}

\item  Set \textsf{Columns} to 2, and and set \textsf{Rows} to 10.

\item  Under \textsf{Fill With}, choose \textsf{Defined By Function}.

\item  In the input box for function name, enter $h$.

\item  Choose \textsf{OK}.
\end{enumerate}

The result is the matrix below on the left. To get the matrix on the right,
choose \textsl{Evaluate Numerically}.

\begin{center}
$\left[ 
\begin{array}{cc}
\frac{1}{100} & \sin \frac{1}{100} \\ 
\frac{1}{50} & \sin \frac{1}{50} \\ 
\frac{3}{100} & \sin \frac{3}{100} \\ 
\frac{1}{25} & \sin \frac{1}{25} \\ 
\frac{1}{20} & \sin \frac{1}{20} \\ 
\frac{3}{50} & \sin \frac{3}{50} \\ 
\frac{7}{100} & \sin \frac{7}{100} \\ 
\frac{2}{25} & \sin \frac{2}{25} \\ 
\frac{9}{100} & \sin \frac{9}{100} \\ 
\frac{1}{10} & \sin \frac{1}{10}
\end{array}
\allowbreak \right] =\allowbreak \left[ 
\begin{array}{cc}
.0\,\allowbreak 1 & 9.\,\allowbreak 999\,8\times 10^{-3} \\ 
.0\,\allowbreak 2 & 1.\,\allowbreak 999\,9\times 10^{-2} \\ 
.0\,\allowbreak 3 & 2.\,\allowbreak 999\,6\times 10^{-2} \\ 
.0\,\allowbreak 4 & 3.\,\allowbreak 998\,9\times 10^{-2} \\ 
.0\,\allowbreak 5 & 4.\,\allowbreak 997\,9\times 10^{-2} \\ 
.0\,\allowbreak 6 & 5.\,\allowbreak 996\,4\times 10^{-2} \\ 
.0\,\allowbreak 7 & 6.\,\allowbreak 994\,3\times 10^{-2} \\ 
.0\,\allowbreak 8 & 7.\,\allowbreak 991\,5\times 10^{-2} \\ 
.0\,\allowbreak 9 & 8.\,\allowbreak 987\,9\times 10^{-2} \\ 
.\,\allowbreak 1 & 9.\,\allowbreak 983\,3\times 10^{-2}
\end{array}
\right] \allowbreak $
\end{center}

\strut 

$\blacktriangleright $ \textbf{To generate a table of values by
concatenating matrices}

\begin{enumerate}
\item[1.]  Click \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{matrix.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Matrix.wmf';file-properties "XNPEU";}} (or
choose \textsf{Insert + Matrix})

\item[2.]  Choose one column and any number of rows.

\item[3.]  Fill in domain values in the rows.

\item[4.]  Put parentheses around the matrix.

\item[5.]  Evaluate \ $f($your matrix$)$.\medskip 
\end{enumerate}

It is also useful to examine the graph of $y=\frac{\sin x}{x}$ on an
interval containing $0$. The graph gives additional strong evidence that $%
\lim_{x\rightarrow 0}\frac{\sin x}{x}=\allowbreak 1$.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Rectangular}
\end{quote}

\begin{quotation}
$\dfrac{\sin x}{x}$
\end{quotation}

\FRAME{dtbpFU}{3in}{2.0003in}{0pt}{\Qcb{$y=\frac{\sin x}{x}$}}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height
2.0003in;depth 0pt;display "PICT";function \TEXUX{$\dfrac{\sin
x}{x}$};linecolor "red";linestyle 1;linethickness 2;pointstyle "point";xmin
"-5";xmax "5";xviewmin "-5.2";xviewmax "5.204";yviewmin "-0.2416";yviewmax
"1.025";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle
"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

\strut

\subsection{\ Special Limits\label{Special limits}}

\textsl{Scientific Notebook} also handles one-sided limits\label%
{One-sided limits} 
\index{Limits!one-sided}, \label{Limits at infinity} 
\index{Limits!at infinity}limits at infinity, and \label{Infinite limits} 
\index{Limits!infinite}infinite limits.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\lim\limits_{x\rightarrow \infty }%
\dfrac{\sqrt{x^{2}+3x+5}}{\sqrt{16x^{2}+5x-3}}=\allowbreak \dfrac{1}{4}$

$\lim\limits_{x\rightarrow 0^{-}}\dfrac{x}{\left| x\right| }=\allowbreak -1$

$\lim\limits_{x\rightarrow 0^{+}}\dfrac{x}{\left| x\right| }=\allowbreak 1$

$\lim\limits_{x\rightarrow 2}\dfrac{x+2}{x-2}=\allowbreak \func{undefined}$

$\lim\limits_{x\rightarrow 2^{+}}\dfrac{x+2}{x-2}=\allowbreak \infty $

$\lim\limits_{x\rightarrow 2^{-}}\dfrac{x+2}{x-2}=\allowbreak -\infty $

$\lim\limits_{x\rightarrow 0}\sin \left( \dfrac{1}{x}\right) =\allowbreak
-1..1\bigskip $
\end{quotation}

\strut

The notation $-1..1$ used in the last limit indicates the interval $-1\,\leq
\,x\,\leq \,1$. Although this output is somewhat unusual, it is explained by
the graph of $y=\sin \frac{1}{x}$. As $x$ approaches $0$, the function
values of $\sin \frac{1}{x}$ include every number in the interval $-1\,\leq
\,x\,\leq \,1$ infinitely often.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Rectangular}
\end{quote}

\begin{quotation}
$\sin \frac{1}{x}$
\end{quotation}

\FRAME{dtbpFU}{3in}{2.0003in}{0pt}{\Qcb{$y=\sin \frac{1}{x}$}}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height
2.0003in;depth 0pt;display "PICT";function \TEXUX{$\sin
\frac{1}{x}$};linecolor "navy";linestyle 1;linethickness 1;pointstyle
"point";xmin "-0.15";xmax "0.15";xviewmin "-0.156";xviewmax
"0.1561";yviewmin "-1.04";yviewmax "1.041";rangeset"X";phi 45;theta
45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name
\TEXUX{$x$};}}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Evaluating Calculus Expressions}{}{}{DM7-1.tex#Calculus
expressions}

\item  \hyperref{Differentiation}{}{}{DM7-3.tex#Derivative}

\item  \hyperref{Indefinite Integration}{}{}{DM7-4.tex#Indefinite integral}

\item  \hyperref{Definite Integrals}{}{}{DM7-6.tex#Definite integral}

\item  \hyperref{Exercises and Solutions}{}{}{DM7-9.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index} {\small Index entries: \hyperref{Limits}{}{}{%
DMIndex.tex#Limits}, \hyperref{Replacing a selection}{}{}{%
DMIndex.tex#Replacing a selection}, \hyperref{Substitution}{}{}{%
DMIndex.tex#Substitution}, \hyperref{Table of values}{}{}{DMIndex.tex#Table
of values}}

\end{document}
