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\begin_body

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%MODULE 1: Epsilon-Delta limits
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%USER DEFINED COMMANDS
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\begin_inset FormulaMacro
\newcommand{\ds}{{\displaystyle }}
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%INPUT THE INSTRUCTIONS PAGE 
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%BUILD  TITLE PAGE 
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\end_layout

\begin_layout Title
A precise notion of limits
\end_layout

\begin_layout Subtitle
We move from the 
\emph on
intuitive
\emph default
 notion of a limit (which until now has been numerical and/or geometric)
 to the 
\emph on
precise
\emph default
 notion of a limit (which is logical and algebraic).
 
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\begin_layout BeginFrame
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{
\end_layout

\end_inset

A Sample problem
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Let 
\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle {\ec f\left(x\right)=\frac{x^{2}-x}{x}}}$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle {\ec w\left(x\right)=\sin\left(\frac{1}{x}\right)}}$
\end_inset

.
 
\end_layout

\begin_layout Standard
Our goal is to find a 
\emph on
precise
\emph default
 way to explain our intuitions that 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0}f(x)=-1}\mathrm{\ but\ }{\ec\lim_{x\rightarrow0}w(x)}\mathrm{\ does\ not\ exist.}
\]

\end_inset


\end_layout

\begin_layout BeginFrame
SAGE worksheets
\end_layout

\begin_layout Standard
\noindent
In this lab you will need the SAGElet 
\begin_inset Quotes eld
\end_inset

Limits: 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset


\begin_inset Quotes erd
\end_inset

.
 Open it in SAGE.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Setup Limits: 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
First we will set up the applet.
 When you first start it, you should see something like this:
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/worksheet_startup_small.png
	lyxscale 30
	scale 20

\end_inset

 
\end_layout

\begin_layout BeginFrame
Setup Limits: 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset


\end_layout

\begin_layout Standard
\noindent
We will set up the applet to work with 
\begin_inset Formula $\ec f(x)$
\end_inset

 first.
 Make the following assignments: 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f\rightarrow(x^{2}-x)/x$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec a\rightarrow0$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec L\rightarrow-1$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\delta\rightarrow0.1$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\epsilon\rightarrow0.1$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec x_{\mathrm{m}in}\rightarrow-1$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec x_{\mathrm{m}ax}\rightarrow1$
\end_inset

 
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

The precise definition of a limit
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
We want to express the idea that as 
\begin_inset Formula $\ec x$
\end_inset

 draws near to 
\begin_inset Formula $\ec a$
\end_inset

, the 
\begin_inset Formula $y$
\end_inset

 values 
\begin_inset Formula $\ec f(x)$
\end_inset

 draw so near to 
\begin_inset Formula $\ec L$
\end_inset

 that you cannot find any 
\emph on
meaningful
\emph default
 distance between them.
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Stated precisely,
\end_layout

\begin_layout Definition
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=L$
\end_inset

 if 
\end_layout

\begin_deeper
\begin_layout Itemize
for any measurable distance 
\begin_inset Formula $\ec\epsilon$
\end_inset

, 
\end_layout

\begin_layout Itemize
we can find a neighborhood around 
\begin_inset Formula $\ec x=a$
\end_inset

 
\end_layout

\begin_layout Itemize
such that for any 
\begin_inset Formula $x$
\end_inset

 value in that neighborhood (except maybe 
\begin_inset Formula $\ec x=a$
\end_inset

), 
\end_layout

\begin_layout Itemize
the distance between the 
\begin_inset Formula $y$
\end_inset

 value of 
\begin_inset Formula $\ec f$
\end_inset

 and 
\begin_inset Formula $\ec L$
\end_inset

 is less than 
\begin_inset Formula $\ec\epsilon$
\end_inset

.
 
\end_layout

\end_deeper
\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Written more precisely,
\end_layout

\begin_layout Definition
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=L$
\end_inset

 if 
\end_layout

\begin_deeper
\begin_layout Itemize
for all 
\begin_inset Formula $\ec\epsilon>0$
\end_inset

,
\begin_inset Newline newline
\end_inset

(
\begin_inset Quotes eld
\end_inset

any measurable distance 
\begin_inset Formula $\ec\epsilon$
\end_inset


\begin_inset Quotes erd
\end_inset

) 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec\delta>0$
\end_inset


\begin_inset Newline newline
\end_inset

(
\begin_inset Quotes eld
\end_inset

a neighborhood around 
\begin_inset Formula $\ec x=a$
\end_inset


\begin_inset Quotes erd
\end_inset

) 
\end_layout

\begin_layout Itemize
such that for all 
\begin_inset Formula $\ec x\in(a-\delta,a+\delta)$
\end_inset

 (except maybe 
\begin_inset Formula $\ec x=a$
\end_inset

),
\begin_inset Newline newline
\end_inset

(
\begin_inset Quotes eld
\end_inset

any 
\begin_inset Formula $x$
\end_inset

 value in that neighborhood
\begin_inset Quotes erd
\end_inset

) 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec|f(x)-L|<\epsilon$
\end_inset

.
\begin_inset Newline newline
\end_inset

(
\begin_inset Quotes eld
\end_inset

the distance between the 
\begin_inset Formula $y$
\end_inset

 values is less than 
\begin_inset Formula $\ec\epsilon$
\end_inset


\begin_inset Quotes erd
\end_inset

) 
\end_layout

\end_deeper
\begin_layout Standard
Switch back and forth between these two slides to see how the words translate
 into symbols.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
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\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Rewritten slightly, we have
\end_layout

\begin_layout Definition
\begin_inset ERT
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\begin_layout Plain Layout

[
\end_layout

\end_inset

(Precise definition of the limit)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=L$
\end_inset

 if 
\end_layout

\begin_deeper
\begin_layout Itemize
for all 
\begin_inset Formula $\ec\epsilon>0$
\end_inset

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec\delta>0$
\end_inset

 
\end_layout

\begin_layout Itemize
such that for all 
\begin_inset Formula $\ec x$
\end_inset

 satisfying 
\begin_inset Formula $\ec|x-a|<\delta$
\end_inset

 (except maybe 
\begin_inset Formula $\ec x=a$
\end_inset

), 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec|f(x)-L|<\epsilon$
\end_inset

.
 
\end_layout

\end_deeper
\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Example with 
\begin_inset Formula $f(x)$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
We illustrate this precise definition of a limit using 
\begin_inset Formula $f(x)$
\end_inset

.
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/good_outermost.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
Currently, SAGE should show the output above.
 Right now we set the precision to 
\begin_inset Formula $\ec\epsilon=0.1$
\end_inset

.
 The goal is that 
\begin_inset Formula $\ec|f(x)-L|<\epsilon$
\end_inset

, or that 
\emph on
the distance between the 
\begin_inset Formula $y$
\end_inset

 values of 
\begin_inset Formula $\ec f$
\end_inset

 and 
\begin_inset Formula $L$
\end_inset

 is smaller than 
\begin_inset Formula $\ec\epsilon$
\end_inset

.
\end_layout

\begin_layout Standard
Geometrically, you see that the graph of 
\begin_inset Formula $f(x)$
\end_inset

 does not stray into the pink regions of the plane.
 How does this reflect the definition?
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/good_outermost.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Geometrically, you see that the graph of 
\begin_inset Formula $f(x)$
\end_inset

 does not stray into the pink regions of the plane.
 How does this reflect the definition?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The distance between the 
\begin_inset Formula $y$
\end_inset

 values of points in the pink regions and 
\begin_inset Formula $\ec L$
\end_inset

 is larger than 
\begin_inset Formula $\ec\epsilon$
\end_inset

.
 This would violate the desired outcome of a limit: that 
\begin_inset Formula $y$
\end_inset

 values grow very close to 
\begin_inset Formula $\ec L$
\end_inset

.
\end_layout

\begin_layout Standard
What about smaller values of 
\begin_inset Formula $\ec\epsilon$
\end_inset

, say 
\begin_inset Formula $\ec\epsilon=0.01$
\end_inset

? Change the value of 
\begin_inset Formula $\ec\epsilon$
\end_inset

 in the SAGElet.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/zoom_bad.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Geometrically, you see that the graph of 
\begin_inset Formula $f(x)$
\end_inset

 does not stray into the pink regions of the plane.
 How does this reflect the definition?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
When we make 
\begin_inset Formula $\ec\epsilon$
\end_inset

 smaller, we have a problem: the blue curve of 
\begin_inset Formula $\ec f$
\end_inset

 now passes in the pink regions of the plane.
 For these points, 
\begin_inset Formula $\ec|f(x)-L|\mathbf{>}\epsilon$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/zoom_bad.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Geometrically, you see that the graph of 
\begin_inset Formula $f(x)$
\end_inset

 does not stray into the pink regions of the plane.
 How does this reflect the definition?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
When we make 
\begin_inset Formula $\ec\epsilon$
\end_inset

 smaller, we have a problem: the blue curve of 
\begin_inset Formula $\ec f$
\end_inset

 now passes in the pink regions of the plane.
 For these points, 
\begin_inset Formula $\ec|f(x)-L|{\mathbf{>}}\epsilon$
\end_inset

.
\end_layout

\begin_layout Standard
However, the phrasing of the definition is that 
\begin_inset Quotes eld
\end_inset

there exists 
\begin_inset Formula $\ec\delta$
\end_inset


\begin_inset Quotes erd
\end_inset

.
 Can we find a value of 
\begin_inset Formula $\ec\delta$
\end_inset

 so the curve no longer enters the pink region? Zoom in on the graph.
 Change the max and min 
\begin_inset Formula $x$
\end_inset

 values so that we zoom into 
\begin_inset Formula $\ec[-0.2,0.2]$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/zoom_bad2.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Geometrically, you see that the graph of 
\begin_inset Formula $f(x)$
\end_inset

 does not stray into the pink regions of the plane.
 How does this reflect the definition?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
It looks like we could make it work by choosing a smaller value of 
\begin_inset Formula $\ec\delta$
\end_inset

.
 Let's try 
\begin_inset Formula $\ec\delta=0.01$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/zoom_good.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dims
\end_layout

\end_inset

 Geometrically, you see that the graph of 
\begin_inset Formula $f(x)$
\end_inset

 does not stray into the pink regions of the plane.
 How does this reflect the definition?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
In this case it worked out.
 Notice that we have 
\begin_inset Formula $\ec\delta=\ec\epsilon$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Proving it
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
We can show this rigorously by falling back on the definition.
 We remind you that the definition is 
\end_layout

\begin_layout Itemize
For all 
\begin_inset Formula $\ec\epsilon>0$
\end_inset

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec\delta>0$
\end_inset

 
\end_layout

\begin_layout Itemize
such that for all 
\begin_inset Formula $\ec x$
\end_inset

 satisfying 
\begin_inset Formula $\ec|x-a|<\delta$
\end_inset

 (except maybe 
\begin_inset Formula $\ec x=a$
\end_inset

), 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec|f(x)-L|<\epsilon$
\end_inset

.
 
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We want to rewrite
\begin_inset Formula 
\begin{align*}
\ec|f(x)-L| & {\ec<\epsilon}\phantom{\mathrm{,\ or\ }{\ec\left|x\right|<\epsilon}.}
\end{align*}

\end_inset

 so that it resembles 
\begin_inset Formula $\ec|x-a|<\mathrm{something}$
\end_inset

, or 
\begin_inset Formula $\ec|x-0|<\mathrm{something}$
\end_inset

.
 Then we could just pick 
\begin_inset Formula $\ec\delta=\mathrm{something}$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
That is,
\begin_inset Formula 
\begin{align*}
\dims|f(x)-L| & {\dims<\epsilon}\phantom{\mathrm{,\ or\ }{\ec\left|x\right|<\epsilon}}\\
\ec\left|\frac{x^{2}-x}{x}-(-1)\right| & {\ec<\epsilon}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Substituted 
\begin_inset Formula $f$
\end_inset

 and 
\begin_inset Formula $L$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
That is,
\begin_inset Formula 
\begin{align*}
\dims|f(x)-L| & {\dims<\epsilon}\phantom{\mathrm{,\ or\ }{\ec\left|x\right|<\epsilon}}\\
\dims\left|\frac{x^{2}-x}{x}-(-1)\right| & {\dims<\epsilon}\\
\ec\left|\frac{x(x-1)}{x}+1\right| & {\ec<\epsilon}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Factored the numerator.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
That is,
\begin_inset Formula 
\begin{align*}
\dims|f(x)-L| & {\dims<\epsilon}\\
\dims\left|\frac{x(x-1)}{x}+1\right| & {\dims<\epsilon}\\
\ec\left|(x-1)+1\right| & {\ec<\epsilon}\mathrm{,\ or\ }{\ec\left|x-0\right|<\epsilon}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Division by 
\begin_inset Formula $x$
\end_inset

 when 
\begin_inset Formula $x\neq0$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5->
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
What value of 
\begin_inset Formula $\ec\delta$
\end_inset

 should we choose? We want 
\begin_inset Formula 
\[
{\ec\left|x-0\right|<\delta\quad\Longrightarrow\quad\left|x-0\right|<\epsilon}.
\]

\end_inset

 Choosing 
\begin_inset Formula $\ec\delta=\epsilon$
\end_inset

 makes it easy.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
We saw this in the SAGElet: when 
\begin_inset Formula $\ec\epsilon=.1$
\end_inset

 we chose 
\begin_inset Formula $\ec\delta=.1$
\end_inset

, and when 
\begin_inset Formula $\ec\epsilon=.01$
\end_inset

 we chose 
\begin_inset Formula $\ec\delta=.01$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Smaller 
\begin_inset Formula $\delta$
\end_inset

 works, too!
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Of course you could have chosen a 
\begin_inset Formula $\ec\delta$
\end_inset

 that was even 
\emph on
smaller
\emph default
 than the one required.
 Try changing 
\begin_inset Formula $\ec\delta$
\end_inset

 to .005 in the SAGE worksheet, and see if the curve passes through the pink
 region.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
You should see something like this:
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/zoom_good_smaller_delta.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
So smaller values of delta work just as well.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

General functions
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
It is not easy to determine a correct value of 
\begin_inset Formula $\ec\delta$
\end_inset

 for every function.
 Even 
\begin_inset Quotes eld
\end_inset

simple
\begin_inset Quotes erd
\end_inset

 functions require care.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Suppose 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

.
 How can we show that 
\begin_inset Formula $\ec\lim_{x\rightarrow1}f(x)=1$
\end_inset

?
\end_layout

\begin_layout Standard
Falling back on the definition, let 
\begin_inset Formula $\ec\epsilon$
\end_inset

 be an arbitrary distance.
 We want
\begin_inset Formula 
\[
{\ec|x-a|=|x-1|<\delta\qquad\Longrightarrow\qquad|f(x)-L|=|x^{2}-1|<\epsilon}.
\]

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We'll try again to rewrite the second inequality in the form
\begin_inset Formula 
\[
{\ec|x-a|=|x-1|<\mathrm{something}}.
\]

\end_inset

 Then we can set 
\begin_inset Formula $\ec\delta=\mathrm{something}$
\end_inset

 and satisfy the definition.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
The second inequality can be rewritten as
\begin_inset Formula 
\begin{align*}
\ec-\epsilon & {\ec<x^{2}-1<\epsilon}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Property of the absolute-value inequality 
\begin_inset Formula $<$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
The second inequality can be rewritten as
\begin_inset Formula 
\begin{align*}
\dims-\epsilon & {\dims<x^{2}-1<\epsilon}\\
\ec1-\epsilon & {\ec<x^{2}<1+\epsilon}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Isolated 
\begin_inset Formula $x^{2}$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
The second inequality can be rewritten as
\begin_inset Formula 
\begin{align*}
\dims1-\epsilon & {\dims<x^{2}<1+\epsilon}\\
\ec\sqrt{1-\epsilon} & {\ec<x<\sqrt{1+\epsilon}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Took square root of both sides.
\begin_inset Newline newline
\end_inset

 In a 
\begin_inset Quotes eld
\end_inset

small enough
\begin_inset Quotes erd
\end_inset

 neighborhood of 
\begin_inset Formula $x=a=1$
\end_inset

,
\begin_inset Newline newline
\end_inset

 
\begin_inset Formula $x$
\end_inset

 is positive, so we don't need 
\begin_inset Formula $\pm\sqrt{}$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<7>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
The second inequality can be rewritten as
\begin_inset Formula 
\begin{align*}
\dims\sqrt{1-\epsilon} & {\dims<x<\sqrt{1+\epsilon}}\\
\ec\sqrt{1-\epsilon}-1 & {\ec<x-1<\sqrt{1+\epsilon}-1}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Subtracted 
\begin_inset Formula $1$
\end_inset

 from all sides.
 Why?
\begin_inset Newline newline
\end_inset

 To find a value for 
\begin_inset Formula $\delta$
\end_inset

 we need the form 
\begin_inset Formula $|x-1|$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<8>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
 The second inequality can be rewritten as
\begin_inset Formula 
\begin{align*}
\dims\sqrt{1-\epsilon} & {\dims<x<\sqrt{1+\epsilon}}\\
\ec\sqrt{1-\epsilon}-1 & {\ec<x-1<\sqrt{1+\epsilon}-1}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
\noindent
Which value gives a 
\emph on
smaller
\emph default
 neighborhood around 
\begin_inset Formula $\ec x=a=1$
\end_inset

? Some algebra shows that 
\begin_inset Formula $\ec\sqrt{1+\epsilon}-1$
\end_inset

 (we skip that step here).
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<9>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
The second inequality can be rewritten as
\begin_inset Formula 
\begin{align*}
\dims\sqrt{1-\epsilon} & {\dims<x<\sqrt{1+\epsilon}}\\
\ec\sqrt{1-\epsilon}-1 & {\ec<x-1<\sqrt{1+\epsilon}-1}.
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
\noindent
Try this in the SAGE applet with a few values of 
\begin_inset Formula $\ec\epsilon$
\end_inset

, and verify that 
\begin_inset Formula $\ec\delta=\sqrt{1+\epsilon}-1$
\end_inset

 keeps the blue function out of the pink regions.
\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

What about 
\begin_inset Formula $w(x)$
\end_inset

?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Now consider 
\begin_inset Formula $\ec w(x)=\sin(1/x)$
\end_inset

.
 To help see why this has no limit at 
\begin_inset Formula $\ec x=0$
\end_inset

, make the following changes to the SAGE applet: 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f\rightarrow sin(1/x)$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec a\rightarrow0$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec L\rightarrow0$
\end_inset

 (we'll start with a guess that the limit is zero) 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\delta\rightarrow0.1$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\epsilon\rightarrow0.1$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec x_{\mathrm{m}in}\rightarrow-1$
\end_inset

 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec x_{\mathrm{m}ax}\rightarrow1$
\end_inset

 
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

What about 
\begin_inset Formula $w(x)$
\end_inset

?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Your initial graph should look like this:
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/wiggle_bad1.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
Notice the intense 
\begin_inset Quotes eld
\end_inset

wiggling
\begin_inset Quotes erd
\end_inset

 as 
\begin_inset Formula $\ec x\rightarrow0$
\end_inset

.
 The blue curve passes through the pink regions many, many times, so this
 value of 
\begin_inset Formula $\ec\delta$
\end_inset

 is too large for 
\begin_inset Formula $\ec\epsilon=.1$
\end_inset

; let's try a smaller value of 
\begin_inset Formula $\ec\delta$
\end_inset

, say 
\begin_inset Formula $\ec\delta=.01$
\end_inset

.
 Be sure to zoom in the 
\begin_inset Formula $x$
\end_inset

 values as well to 
\begin_inset Formula $\ec[-.05,.05]$
\end_inset

.
\end_layout

\begin_layout BeginFrame
What about 
\begin_inset Formula $w(x)$
\end_inset

?
\end_layout

\begin_layout Standard
\noindent
Now you should see this (or something similar):
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/wiggle_bad2.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
The wiggling has not ceased to take us into the pink zone, so 
\begin_inset Formula $\ec\delta$
\end_inset

 is still too large for this value of 
\begin_inset Formula $\ec\epsilon$
\end_inset

.
 Let's try one more smaller value of 
\begin_inset Formula $\ec\delta$
\end_inset

.
\end_layout

\begin_layout BeginFrame
What about 
\begin_inset Formula $w(x)$
\end_inset

?
\end_layout

\begin_layout Standard
\noindent
Now you should see this (or something similar):
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/wiggle_bad3.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
No wonder mathematicians call this the 
\emph on
Infinite Wiggle
\emph default
.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

What is going on?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
No matter how small we made 
\begin_inset Formula $\ec\delta$
\end_inset

, the graph of 
\begin_inset Formula $\ec w(x)$
\end_inset

 only wiggled all the more intensely.
 We need to convince ourselves of this.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/wiggle_bad3.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
Notice that the curve touches 
\begin_inset Formula $\ec y=1$
\end_inset

 many, many times.
 What 
\begin_inset Formula $x$
\end_inset

 values give 
\begin_inset Formula $\ec\sin(x)=1$
\end_inset

?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
You should know that 
\begin_inset Formula $\ec x=\frac{\pi}{2},-\frac{\pi}{2},\frac{5\pi}{2},-\frac{5\pi}{2},\ldots$
\end_inset

 produce 
\begin_inset Formula $\ec\sin(x)=1$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
We are dealing with the function 
\begin_inset Formula $\ec\sin(1/x)$
\end_inset

, so look at the reciprocal of these numbers: 
\begin_inset Formula $\ec x=\frac{2}{\pi},-\frac{2}{\pi},\frac{2}{5\pi},-\frac{2}{5\pi},\ldots$
\end_inset

 produce 
\begin_inset Formula $\ec\sin(1/x)=1$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
These values approach zero: 
\begin_inset Formula $\ec\frac{2}{\pi},\frac{2}{5\pi},\frac{2}{9\pi},\ldots$
\end_inset

.
 The pattern goes on forever, 
\emph on
all
\emph default
 their 
\begin_inset Formula $y$
\end_inset

 values are 1, and the limit of these numbers is 0.
\end_layout

\begin_layout BeginFrame
What is going on?
\end_layout

\begin_layout Standard
\noindent
No matter how small we made 
\begin_inset Formula $\delta$
\end_inset

, the graph of 
\begin_inset Formula $w(x)$
\end_inset

 only wiggled all the more intensely.
 We need to convince ourselves of this.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename figures/wiggle_bad3.png
	lyxscale 30
	scale 30

\end_inset

 
\end_layout

\begin_layout Standard
The upshot? 
\end_layout

\begin_layout Itemize
Regardless of the value of 
\begin_inset Formula $\ec\delta$
\end_inset

 
\end_layout

\begin_layout Itemize
we can find 
\begin_inset Formula $\ec x$
\end_inset

 
\end_layout

\begin_layout Itemize
such that 
\begin_inset Formula $\ec|x-0|<\delta$
\end_inset

 
\end_layout

\begin_layout Itemize
but 
\begin_inset Formula $\ec|f(x)-0|=1>0.1=\epsilon!$
\end_inset

 
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Thus...
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We conclude that 
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow0}\sin\left(\frac{1}{x}\right)
\]

\end_inset

 
\series bold
does not exist
\series default
.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Conclusion
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
You have seen 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 the precise definition of a limit
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

:
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setlength{
\backslash
shadeboxwidth}{4in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Definitions
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=L$
\end_inset

 if 
\end_layout

\begin_deeper
\begin_layout Itemize
for all 
\begin_inset Formula $\ec\epsilon>0$
\end_inset

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec\delta>0$
\end_inset

 
\end_layout

\begin_layout Itemize
such that for all 
\begin_inset Formula $\ec x$
\end_inset

 satisfying 
\begin_inset Formula $\ec|x-a|<\delta$
\end_inset

 (except maybe 
\begin_inset Formula $\ec x=a$
\end_inset

) 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec|f(x)-L|<\epsilon$
\end_inset

.
\end_layout

\end_deeper
\end_deeper
\begin_layout Itemize
You have had an introduction to finding 
\begin_inset Formula $\ec\delta$
\end_inset

 algebraically.
 It takes a lot of practice to grow accustomed to it.
 
\end_layout

\begin_layout Itemize
Why does this give us the precise definition of the limit? 
\end_layout

\begin_deeper
\begin_layout Itemize
No matter how close (
\begin_inset Formula $\ec\epsilon$
\end_inset

) we check near 
\begin_inset Formula $\ec y=L$
\end_inset

, 
\end_layout

\begin_layout Itemize
we can find a neighborhood (
\begin_inset Formula $\ec\delta$
\end_inset

) of 
\begin_inset Formula $\ec x=a$
\end_inset

 
\end_layout

\begin_layout Itemize
such that the distance between 
\begin_inset Formula $\ec L$
\end_inset

 and 
\begin_inset Formula $\ec f(x)$
\end_inset

 is less than 
\begin_inset Formula $\ec\epsilon$
\end_inset

 (
\begin_inset Formula $\ec|f(x)-L|<\epsilon$
\end_inset

).
 
\end_layout

\begin_layout Itemize
Since 
\begin_inset Formula $\ec\epsilon$
\end_inset

 is 
\emph on
any
\emph default
 measurable distance, the limit of 
\begin_inset Formula $\ec f(x)$
\end_inset

 is 
\begin_inset Formula $\ec L$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Itemize
You can use the SAGElet 
\begin_inset Quotes eld
\end_inset

Limits: 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset


\begin_inset Quotes erd
\end_inset

 to illustrate this, both when the limit exists and when it does not.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
