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\begin{document}
%MODULE 1: Epsilon-Delta limits


%USER DEFINED COMMANDS
\global\long\def\ds{{\displaystyle }}
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%INPUT THE INSTRUCTIONS PAGE 


%BUILD  TITLE PAGE 



\title{An intuitive notion of limits}


\subtitle{In this lesson we introduce the \emph{fundamental} concept of Calculus:
\emph{\hilite the limit}. For now we will take a more intuitive approach,
looking at \emph{graphical} and \emph{numerical} approaches to the
limit. We will introduce a rigorous, \emph{algebraic} approach to
the limit in another module.}

\makebeamertitle

\lyxframeend{}\lyxframe{{Two sample problems}}

\noindent Let 
\begin{itemize}
\item $\ec{\displaystyle {\ec f\left(x\right)=\frac{x^{2}-x}{x}}}$, and 
\item $\ec{\displaystyle {\ec w\left(x\right)=\sin\left(\frac{1}{x}\right)}}$. 
\end{itemize}
Both of these functions are undefined \emph{at} $\ec x=0$. We want
to explore how they behave as $\ec x$ \emph{approaches} $\ec0$.


\lyxframeend{}\lyxframe{{The limit}}
\begin{definition}%{}
[(Intuitive definition of a limit)]The {\hilite limit} indicates
the value to which a variable or a function grows arbitrarily close.
We write $\ec\lim_{x\rightarrow a}f(x)=L$ to mean that the $y$ values
of $\ec f(x)$ grow arbitrarily close to $\ec L$ when the $x$ values
grow arbitrarily close to $\ec a$.
\end{definition}%{}
\noindent In the case examples listed on the previous page, we want
to know the \emph{limit} of $\ec f$ or $\ec w$ as $\ec x$ approaches
$0$.

In general, when we want to know the \emph{limit} of $\ec f$ as $\ec x$
approaches some value $\ec a$, we write
\[
{\ec\lim_{x\rightarrow a}f(x)}.
\]
In this worksheet, we want to evaluate
\[
{\ec\lim_{x\rightarrow0}f(x)}\quad\mathrm{and}\quad{\ec\lim_{x\rightarrow0}w(x)}.
\]



\lyxframeend{}\lyxframe{{SAGE worksheets}}

\noindent In this lab you will not need a special SAGE worksheet,
but you will need to have a blank SAGE worksheet handy, so go ahead
and login to SAGE and open a new worksheet.


\lyxframeend{}\lyxframe{{The numerical approach}}
\begin{overlayarea}
{\textwidth}{6in}

\noindent The first method to evaluating a limit is called the \emph{numerical
approach}. To estimate the limit of $\ec f$ as $\ec x$ approaches
$\ec0$, we build a table of $x$-$y$ values using values of $\ec x$
that get close to $0$. We should choose values that lie on both sides
of $0$, in order to make sure the limits are the same from left \emph{and}
right.
\begin{onlyenv}%{}
<1>

First we build the table:


{\ec 


\begin{center}
\begin{tabular}{c|ccccccc}
$x$  & -1  & -0.1  & -0.01  & {\hilite 0}  & 0.01  & 0.1  & 1 \tabularnewline
\hline 
$y$  &  &  &  & {\hilite ???}  &  &  & \tabularnewline
\end{tabular}
\par\end{center}


} \begin{explanation}Recall that $y=f(x)=\frac{x^{2}-x}{x}$.\end{explanation}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<2->

Now we fill in the table:


\begin{center}
{\ec %
\begin{tabular}{c|ccccccc}
$x$  & -1  & -0.1  & -0.01  & {\hilite 0}  & 0.01  & 0.1  & 1 \tabularnewline
\hline 
$y$  & -2  & -1.1  & -1.01  & {\hilite ???}  & -0.99  & -0.9  & 0 \tabularnewline
\end{tabular}}
\par\end{center}

\end{onlyenv}%{}


\begin{onlyenv}%{}
<2>

\begin{center}
Can you guess the value of $\ec\lim_{x\rightarrow0}f(x)$?
\par\end{center}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>

It \emph{looks} as if the $y$-values are approaching $0$. This suggests
that
\[
{\ec\lim_{x\rightarrow0}f(x)=-1}.
\]


\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{SAGE and numerical computations}}
\begin{overlayarea}
{\textwidth}{6in}

\noindent You have several options for creating the table below.


{\ec 


\begin{center}
\begin{tabular}{c|ccccccc}
$x$  & -1  & -0.1  & -0.01  & {\hilite 0}  & 0.01  & 0.1  & 1 \tabularnewline
\hline 
$y$  & -2  & -1.1  & -1.01  & {\hilite ???}  & -0.99  & -0.9  & 0 \tabularnewline
\end{tabular}
\par\end{center}


}
\begin{itemize}
\item You can find these values by hand. This function isn't very difficult,
so that's easy. \end{itemize}
\begin{onlyenv}%{}
<1>
\begin{itemize}
\item You can use a calculator. This can be rather tiresome on the fingers
when the function is involved, but it probably saves times from hand
computation. (If your calculator has a table feature, your fingers
won't be so tired.) 
\end{itemize}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<2>
\begin{itemize}
\item You can also use SAGE. How? 
\end{itemize}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>
\begin{itemize}
\item You can also use SAGE. How? 

\begin{itemize}
\item The simplest way is to define the function $\ec f$ by typing \texttt{\ttcolor
f = (x\^{}2-x)/x}. To evaluate it at $\ec x=-1$, type \texttt{\ttcolor
f(-1)}. Repeat this for other values. 
\end{itemize}
\end{itemize}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<4>
\begin{itemize}
\item You can also use SAGE. How? 

\begin{itemize}
\item The simplest way is to define the function $\ec f$ by typing \texttt{\ttcolor
f = (x\^{}2-x)/x}. To evaluate it at $\ec x=-1$, type \texttt{\ttcolor
f(-1)}. Repeat this for other values. 
\item A somewhat faster method is to use a {\hilite loop}. You can do
this by typing {\ttcolor {[}f(a) for a in {[}-1,-0.1,-0.01,0.01,0.1,1{]}{]}}.
(\emph{Try it!}) In this {\ttcolor for} loop, the variable {\ttcolor
a} takes all the values specified in the list, and the expression
{\ttcolor f(a)} is evaluated for each of those values. 
\end{itemize}
\end{itemize}
\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{The graphical approach}}

\noindent Now let's apply the graphical approach to $\ec f(x)$. Sketch
a graph of $\ec f$ around $\ec x=0$. You should see something like
this: 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_f} 
\end{figure}



\lyxframeend{}\lyxframe{The graphical approach: WARNING}

\noindent {\dims Now let's apply the graphical approach to $f(x)$.
Sketch a graph of $f$ around $x=0$. You should see something like
this:} 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_f} 
\end{figure}


\noindent You have to be careful reading plots that are generated
by a computer. The computer doesn't know how to discover the division
by zero, so it draws the curve as if there were a point at $\ec(0,-1)$
\emph{even though there is no point there}.


\lyxframeend{}\lyxframe{The graphical approach: the limit}

\noindent {\dims Now let's apply the graphical approach to $f(x)$.
Sketch a graph of $f$ around $x=0$. You should see something like
this:} 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.0625]{figures/plot_f_twosided} 
\end{figure}


\noindent It is often easy to {}``see'' the limit from a graph.
Recall that $\ec\lim_{x\rightarrow0}f(x)$ is the value that $\ec f$
approaches as $\ec x$ approaches $\ec0$. Imagine that you are an
ant walking along the graph of $\ec f$. What $y$ value would you
approach when your $x$ value approaches $0$? That point is the \emph{limit}
of $\ec f$ as $\ec x$ approaches $0$. Here 
\[
\ec{\lim_{x\rightarrow0}f(x)=-1}.
\]



\lyxframeend{}\lyxframe{{SAGE and plots}}

\noindent As with the numerical approach, you have several options
for sketching a graph. 
\begin{itemize}
\item You can sketch it by hand using an $x$-$y$ table. This takes a long
time, and is subject to human error. 
\item You can use a graphing calculator. This doesn't take very long, but
requires a \$100~calculator that isn't very flexible in its graphing. 
\item You can use SAGE's {\ttcolor plot()} function. This doesn't take
very long, does not require any money besides what you've already
paid for the class, and is very flexible. See the module {}``An Introduction
to SAGE'' for more details. 
\end{itemize}

\lyxframeend{}\lyxframe{{Specificity of a limit}}
\begin{overlayarea}
{\textwidth}{6in}

\noindent Notice that we have talked about computing or looking from
\emph{both} sides. The limit can have only \emph{one specific} value.
It should be the same whether $\ec x$ approaches $\ec a$ from negative
values ({}``from the left'') or whether $\ec x$ approaches $\ec a$
from positive values ({}``from the right'').
\begin{onlyenv}%{}
<1>

\noindent We write this property as 
\[
\ec\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)
\]
 where $\ec x\rightarrow a^{-}$ reads {}``as $\ec x$ approaches
$\ec a$ from the left'' and $\ec x\rightarrow a^{+}$ reads {}``as
$\ec x$ approaches $\ec a$ from the right''.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<2>

Both the table of $x$-$y$ values and the graph suggested that 
\[
{\ec\lim_{x\rightarrow0^{-}}f(x)=\lim_{x\rightarrow0^{+}}f(x)},
\]
 so we were able to conclude that $\ec{\displaystyle \lim_{x\rightarrow0}f(x)=0}$;
that is, the \emph{two-sided limit} exists.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>

On the other hand, the function $\ec g$ graphed below has \emph{no}
limit at $\ec x=0$, because the limits from the left and right disagree.
\begin{figure}[h!]
 

\begin{centering}
\includegraphics[scale=0.2]{figures/twosided_necessary} 
\par\end{centering}

\caption{ ${\displaystyle \lim_{x\rightarrow0^{-}}g(x)\neq\lim_{x\rightarrow0^{+}}g(x)}$}
\end{figure}


\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{Second example}}

\noindent Now let's try to estimate
\[
\ec\lim_{x\rightarrow0}w(x)
\]
where
\[
{\ec w(x)=\sin\left(\frac{1}{x}\right)}.
\]



\lyxframeend{}\lyxframe{{Second example: The numerical approach}}
\begin{overlayarea}
{\textwidth}{6in}

\noindent To estimate the limit of $\ec w$ as $\ec x$ approaches
$0$, we build a table of $x$-$y$ values using values of $\ec x$
that get close to $\ec0$.
\begin{onlyenv}%{}
<1>

Fill in the table, rounding to the ten-thousandths place.


\begin{center}
{\ec%
\begin{tabular}{c|ccccccc}
$x$  & -1.0  & -0.1  & -0.01  & {\hilite 0}  & 0.01  & 0.1  & 1.0 \tabularnewline
\hline 
$y$  &  &  &  & {\hilite ???}  &  &  & \tabularnewline
\end{tabular}}\\
 
\par\end{center}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<2>

Fill in the table, rounding to the ten-thousandths place.


\begin{center}
{\ec%
\begin{tabular}{c|ccccccc}
$x$  & -1.0  & -0.1  & -0.01  & {\hilite 0}  & 0.01  & 0.1  & 1.0 \tabularnewline
\hline 
$y$  & -0.8415  & 0.5540  & 0.5063  & {\hilite ???}  & -0.5063  & -0.5440  & 0.8415 \tabularnewline
\end{tabular}}\\
 Can you guess the value of $\ec\lim_{x\rightarrow0}w(x)$? 
\par\end{center}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>

Fill in the table, rounding to the ten-thousandths place.


\begin{center}
{\ec%
\begin{tabular}{c|ccccccc}
$x$  & -1.0  & -0.1  & -0.01  & {\hilite 0}  & 0.01  & 0.1  & 1.0 \tabularnewline
\hline 
$y$  & -0.8415  & -0.5540  & -0.5063  & {\hilite 0.5?}  & 0.5063  & 0.5440  & 0.8415 \tabularnewline
\end{tabular}}
\par\end{center}


You might try to {}``guess'' from this data that 
\[
{\ec\lim_{x\rightarrow0}w(x)\approx0.5}.
\]
You might also try to {}``guess'' that it should be the average;
that is, 
\[
{\ec\lim_{x\rightarrow0}w(x)=0}.
\]
 \emph{\hilite That would be a very bad guess.}

\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{Second example: The graphical approach}}
\begin{overlayarea}
{\textwidth}{6in}

\noindent The graphical approach will show us why we cannot say that
the limit is either of those values. Sketch the graph of $\ec w(x)$.


\pause{}


You should see the following: 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_w} 
\end{figure}



What is going on?

\end{overlayarea}

\lyxframeend{}\lyxframe{{The infinite wiggle}}

\noindent This graph is called an \emph{infinite wiggle}. As the curve
gets closer to $\ec x=0$, the $y$-values start to vibrate more and
more wildly between $\ec y=-1$ and $\ec y=1$. 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_w} 
\end{figure}



\lyxframeend{}\lyxframe{The infinite wiggle zoomed}
\begin{overlayarea}
{\textwidth}{6in}

\noindent This graph is something called an \emph{infinite wiggle}.
As the curve gets closer to $\ec x=0$, the $y$-values start to vibrate
more and more wildly between $\ec y=-1$ and $\ec y=1$.
\begin{onlyenv}%{}
<1>

\noindent 
\begin{figure}[h!]
\centering{}\includegraphics[scale=0.2]{figures/plot_w_zoomed} 
\end{figure}



\noindent You can zoom in closer if you wish; $\ec w$ wiggles all
the more. It's a regular party down there.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<2>

\noindent 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_w} 
\end{figure}



What then should we say about the limit of $\ec w$ as $\ec x$ approaches
$0$? The function does not approach \emph{one} specific value, so
we say that {\hilite the limit \emph{does not exist}}.

\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{Fixing the numerical approach}}

\noindent We can see this clearly by computing more $x$-$y$ values
in the numerical approach. Here is a slightly larger table of values
approaching from the left:

\begin{center}
{\ec %
\begin{tabular}{c|cccccc}
$x$  & -1.0  & -0.1  & -0.01  & -0.001  & -0.0001  & -0.00001 \tabularnewline
\hline 
$y$  & $0.8415$  & $0.5540$  & $0.5064$  & $-0.8269$  & $0.3056$  & $-0.0357$ \tabularnewline
\end{tabular}}
\par\end{center}

\noindent The numbers they bounce around, and do not appear to approach
any one value. This also provides a clue that the function is not
settling on a limit.


\lyxframeend{}\lyxframe{{Use both methods carefully}}

\noindent To estimate a limit, it is probably a good idea to use both
methods. You should build $x$-$y$ tables with a \emph{lot} of carefully
chosen $\ec x$-values, and you should zoom into your graphs to make
sure that no strange behaviors are disguised by too large a viewing
area. 
\begin{itemize}
\item Sometimes numerical values can appear to bounce around, but a graph
will show that there is a trend for the bouncing to even out as the
$y$-values approach a definite number. For an example of this phenomenon,
look at a plot of the function $\ec x(1+\sin(100x))$ on the interval
$\ec[-1,1]$. 
\item On the other hand, the graph of a function may sometimes suggest a
limit when none exists, whereas a numerical approach will show that
the values are in fact bouncing around. For an example of this phenomenon,
look at a plot of the function $\ec0.01\sin(1/x)+x$ on the interval
$\ec[-1,1]$. 
\end{itemize}

\lyxframeend{}\lyxframe{{Estimation vs. evaluation}}

\noindent All the same, these methods only allow us to \emph{estimate}
a limit, they do not allow us to \emph{evaluate} the limit. We will
develop a precise method to help us evaluate limits \emph{precisely}
in a future module.


\lyxframeend{}\lyxframe{{Conclusion}}
\begin{itemize}
\item The \emph{limit} of a function $\ec f$ as $\ec x$ approaches a value
$\ec a$ is the value $\ec f$ approaches as we close in on $\ec x=a$
from the left and right sides of $\ec a$. 
\item There can only be one value for a limit. Limits do not always exist. 
\item We studied two methods of \emph{estimating} limits: 

\begin{itemize}
\item a \emph{numerical} method, and 
\item a \emph{graphical} method. 
\end{itemize}
\item Using both these methods can give us an idea of what the value of
a limit is, but they do not give us \emph{certainty} as to what the
value of a limit is. 
\end{itemize}

\lyxframeend{}
\end{document}
