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\begin_body

\begin_layout Standard
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%MODULE 1: Epsilon-Delta limits
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%USER DEFINED COMMANDS
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%INPUT THE INSTRUCTIONS PAGE 
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%BUILD  TITLE PAGE 
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\end_layout

\begin_layout Title
Limits at 
\begin_inset Formula $\infty$
\end_inset


\end_layout

\begin_layout Subtitle
We look at the long-term behavior of functions.
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\begin_layout BeginFrame
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{
\end_layout

\end_inset

Introduction
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Up to now we have only considered limits as they approach 
\emph on
a fixed number
\emph default
 
\begin_inset Formula $\ec a$
\end_inset

:
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow a}f(x).
\]

\end_inset

 Often we are interested as well in the 
\emph on
long-term
\emph default
 behavior of a function: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
What happens to the 
\begin_inset Formula $y$
\end_inset

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

 increases without bound? That is, what is
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow\infty}f(x)?
\]

\end_inset

 
\end_layout

\begin_layout Itemize
What happens to the 
\begin_inset Formula $y$
\end_inset

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

 decreases without bound? That is, what is
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow-\infty}f(x)?
\]

\end_inset

 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

Three sample problems
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Let 
\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle {\ec f\left(x\right)=\frac{3x+1}{x}}}$
\end_inset

, 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec s(x)=\sin(x)$
\end_inset

, and 
\end_layout

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

.
 
\end_layout

\begin_layout Standard
\noindent
We want to estimate, or even evaluate,
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow\pm\infty}f(x)}\quad\mathrm{and}\quad{\ec\lim_{x\rightarrow\pm\infty}s(x)}\quad\mathrm{and}\quad{\ec\lim_{x\rightarrow\pm\infty}d(x)},
\]

\end_inset


\emph on
assuming they exist!
\emph default
.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

The limit at infinity
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Recall from previous lessons that: 
\end_layout

\begin_layout Itemize
the mathematical term 
\emph on
limit
\emph default
 indicates the value a variable or a function approaches; 
\end_layout

\begin_layout Itemize
we can use numerical or graphical methods to estimate the value of a limit,
 if it exists; and 
\end_layout

\begin_layout Itemize
we can use the precise definition of a limit (
\begin_inset Formula $\varepsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset

) to show rigorously that our estimate is correct, or that none exists.
 
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Infinity is not a number!
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
It is important to keep in mind that 
\end_layout

\begin_layout Standard
\noindent
\align center

\emph on
infinity is not a number!
\end_layout

\begin_layout Standard
We cannot 
\begin_inset Quotes eld
\end_inset

substitute
\begin_inset Quotes erd
\end_inset

 it in place of 
\begin_inset Formula $\ec x$
\end_inset

, add to it, subtract from it, and so forth.
\end_layout

\begin_layout Standard
Infinity is a 
\emph on
word
\emph default
 that we use as a shorthand for an 
\emph on
idea
\emph default
: namely, increasing or decreasing without bound.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

SAGE worksheets
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

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

\begin_layout BeginFrame
\noindent
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

First example: numerical
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

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\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Let's use all three methods on the first example,
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow\pm\infty}f(x)}\quad\mathrm{where}\quad{\ec f(x)=\frac{3x+1}{x}}.
\]

\end_inset

 We will start with a numerical estimate: what happens to the values of
 
\begin_inset Formula $\ec f$
\end_inset

 as 
\begin_inset Formula $\ec x$
\end_inset

 increases without bound?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Since 
\begin_inset Formula $\ec x$
\end_inset

 increases without bound, we'll build a table of 
\begin_inset Formula $x$
\end_inset

-
\begin_inset Formula $y$
\end_inset

 values where 
\begin_inset Formula $\ec x$
\end_inset

 increases.
 Feel free to pick grotesquely big numbers!
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3-4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
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<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
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\end_layout

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</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

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1 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
100 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1000 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10000 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1,000,000,000 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
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\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\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
\noindent
Remember that you can use SAGE to pass quickly through the computations:
 define a function 
\begin_inset Formula $\ec f$
\end_inset

, then evaluate 
\begin_inset Formula $\ec f$
\end_inset

 at each point.
 You can also use a loop: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 [f(a) for a in [1,10,\SpecialChar \ldots{}
]]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 as described in the module on intuitive limits.
\end_layout

\end_deeper
\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 ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="8">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
100.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1,000,000,000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\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


\begin_inset Formula $\rightarrow\infty$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.1 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.01 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.001 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.0001 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.000000001 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\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


\begin_inset Formula $\rightarrow3$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
If everything goes well, you should see numbers corresponding to the table
 above.
 This suggests that 
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow\infty}f(x)=3.
\]

\end_inset


\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

First example: graphical
\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
Let's use all three methods on the first example,
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow\pm\infty}f(x)\quad\mathrm{where}\quad f(x)=\frac{3x+1}{x}.
\]

\end_inset

Now let's try a graphical approach.
 Plot the graph of 
\begin_inset Formula $\ec f(x)$
\end_inset

 several times, with different maximum values of 
\begin_inset Formula $\ec x$
\end_inset

.
\end_layout

\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_layout Standard
If we try 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 plot(f,-10,10)
\family default
 we see the following: 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_f_too_large.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
From the graph we might guess that
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow\infty}f(x)=0.
\]

\end_inset


\emph on
But this disagrees with our numerical estimate!
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

What happened?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Take a closer look at the graph: 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_f_too_large.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Do you see how the 
\begin_inset Formula $y$
\end_inset

 values have such large magnitude? They go up to 100-ish and down to -1900-ish.
\end_layout

\begin_layout Standard
To get a more accurate picture, we can use the 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 show()
\family default
 command with the 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 ymin
\family default
 and 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 ymax
\family default
 attributes.
 Try typing this:
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ttcolor 
\end_layout

\end_inset

fplot = plot(f,-10,10)
\end_layout

\begin_layout Standard
fplot.show(ymin=-10,ymax=10)
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

First example: graphical
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
This new approach gives us a much better picture: 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_f_near_origin.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
This new plot 
\emph on
suggests
\emph default
 that the 
\begin_inset Formula $y$
\end_inset

 values approach 3.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

First example: graphical
\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
However, our graph isn't that convincing.
 It 
\emph on
suggests
\emph default
 that the 
\begin_inset Formula $y$
\end_inset

 values approach 3.
 But we're only going out to 
\begin_inset Formula $\ec x=10$
\end_inset

.
 Lots of funny things can happen beyond 
\begin_inset Formula $\ec x=10$
\end_inset

.
 To get around this, experiment with the 
\begin_inset Formula $x$
\end_inset

-interval in the plot.
\end_layout

\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
We might, for example, try the interval 
\begin_inset Formula $\ec[1,100]$
\end_inset

: 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_f_0_100.png
	lyxscale 20
	scale 20

\end_inset

 
\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
Or the interval 
\begin_inset Formula $\ec[1,10000]$
\end_inset

: 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_f_0_10000.png
	lyxscale 20
	scale 20

\end_inset

 
\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
You can also shrink the 
\begin_inset Formula $y$
\end_inset

-interval in the 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 show()
\family default
 command to 
\begin_inset Formula $\ec[2.9,3.1]$
\end_inset

: 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_f_small_y_interval.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
(Use 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 ymin=2.9,ymax=3.1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 as options to 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 show()
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\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
No matter how we change the graph, it 
\emph on
appears
\emph default
 that the 
\begin_inset Formula $y$
\end_inset

 values are approaching 
\begin_inset Formula $\ec y=3$
\end_inset

 as 
\begin_inset Formula $\ec x$
\end_inset

 increases without bound.
 Intuitively, we believe that
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow\infty}f(x)=3.
\]

\end_inset

(The same also appears to be true for 
\begin_inset Formula $\ec-\infty$
\end_inset

.)
\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

A precise approach!
\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
How can we 
\emph on
convince
\emph default
 ourselves that a limit at infinity is 
\begin_inset Formula $\ec L$
\end_inset

? We will need a precise definition for the limit at infinity.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2-3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
For any measurable distance 
\begin_inset Formula $\ec\varepsilon$
\end_inset

, we want to know that eventually the 
\begin_inset Formula $y$
\end_inset

 value of 
\begin_inset Formula $\ec f(x)$
\end_inset

 is indistinguishable from the limit 
\begin_inset Formula $\ec3$
\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
In other words, we want 
\end_layout

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

, 
\end_layout

\begin_layout Itemize
sooner or later, 
\emph on
every
\emph default
 
\begin_inset Formula $x$
\end_inset

 value 
\end_layout

\begin_layout Itemize
produces a 
\begin_inset Formula $y$
\end_inset

 value that is closer to 
\begin_inset Formula $\ec L$
\end_inset

 than 
\begin_inset Formula $\ec\varepsilon$
\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

<4->
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Rethinking our wording again,
\end_layout

\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Limit at 
\begin_inset Formula $\infty$
\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

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

 if 
\end_layout

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

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec N$
\end_inset

 such that for 
\emph on
every
\emph default
 
\begin_inset Formula $\ec x>N$
\end_inset

 
\end_layout

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

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Let's try this with the example, using 
\begin_inset Formula $\ec\varepsilon=0.01$
\end_inset

.
 We want
\begin_inset Formula 
\begin{align*}
{\ec\left|f(x)-L\right|} & {\ec<0.01}.
\end{align*}

\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
Let's try this with the example, using 
\begin_inset Formula $\ec\varepsilon=0.01$
\end_inset

.
 We want
\begin_inset Formula 
\begin{align*}
{\dims\left|f(x)-L\right|} & {\dims<0.01}\\
{\ec\left|\frac{3x+1}{x}-3\right|} & {\ec<0.01}.
\end{align*}

\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
Let's try this with the example, using 
\begin_inset Formula $\ec\varepsilon=0.01$
\end_inset

.
 We want
\begin_inset Formula 
\begin{align*}
{\dims\left|\frac{3x+1}{x}-3\right|} & {\dims<0.01}\\
{\ec-0.01<\frac{3x+1}{x}-3} & {\ec<0.01}.
\end{align*}

\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
Let's try this with the example, using 
\begin_inset Formula $\ec\varepsilon=0.01$
\end_inset

.
 We want
\begin_inset Formula 
\begin{align*}
{\dims-0.01<\frac{3x+1}{x}-3} & {\dims<0.01}\\
{\ec3-0.01<\frac{3x+1}{x}} & {\ec<3+0.01}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Added 
\begin_inset Formula $3$
\end_inset

 to both sides.
\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

<9>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Let's try this with the example, using 
\begin_inset Formula $\ec\varepsilon=0.01$
\end_inset

.
 We want
\begin_inset Formula 
\begin{align*}
{\dims3-0.01<\frac{3x+1}{x}} & {\dims<3+0.01}\\
{\ec x(3-0.01)<3x+1} & {\ec<x(3+0.01)}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Multiplied 
\begin_inset Formula $x$
\end_inset

 to both sides.
\begin_inset Newline newline
\end_inset

 Since 
\begin_inset Formula $x>0$
\end_inset

 as 
\begin_inset Formula $x\rightarrow\infty$
\end_inset

,
\begin_inset Newline newline
\end_inset

 the direction of the inequalities does not change.
\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

<10>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Let's try this with the example, using 
\begin_inset Formula $\ec\varepsilon=0.01$
\end_inset

.
 We want
\begin_inset Formula 
\begin{align*}
{\dims x(3-0.01)<3x+1} & {\dims<x(3+0.01)}\\
{\ec-0.01x<1} & {\ec<0.01x}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Subtracted 
\begin_inset Formula $3x$
\end_inset

 from both sides.
\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

<11>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Let's try this with the example, using 
\begin_inset Formula $\ec\varepsilon=0.01$
\end_inset

.
 We want
\begin_inset Formula 
\begin{align*}
{\dims-0.01x<1} & {\dims<0.01x}\\
{\ec-0.01x<1} & \mathrm{\ and\ }{\ec1<0.01x}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Meaning of the inequality.
\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

<12->
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Let's try this with the example, using 
\begin_inset Formula $\ec\varepsilon=0.01$
\end_inset

.
 We want
\begin_inset Formula 
\begin{align*}
{\dims-0.01x<1} & \mathrm{\ and\ }{\dims1<0.01x}\\
\mathrm{so\ we\ want\ }\quad{\ec x>-\frac{1}{0.01}} & \mathrm{\ and\ }{\ec x>\frac{1}{0.01}}.
\end{align*}

\end_inset

 The larger of these is 
\begin_inset Formula $\ec1/0.01=100$
\end_inset

, so we want 
\begin_inset Formula $\ec x>100$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
In fact, 
\begin_inset Formula $\ec f(101)\approx3.0099$
\end_inset

 and 
\begin_inset Formula $\ec\left|3.0099-3\right|=.0099<.01$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Second example
\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
Let's look at 
\begin_inset Formula $\ec s(x)=\sin(x)$
\end_inset

.
\end_layout

\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
If we build a table of values extending towards infinity\SpecialChar \ldots{}

\end_layout

\begin_layout Standard
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
100.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1,000,000,000.0 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.8415 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-0.5540 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-0.5063 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.8269 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-0.3056 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.5458 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\SpecialChar \ldots{}
we don't see any trend to 
\emph on
one
\emph default
 special number.
\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
If we plot the function for intervals with larger and larger 
\begin_inset Formula $x$
\end_inset

 values\SpecialChar \ldots{}
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_s_near_origin.png
	lyxscale 20
	scale 20

\end_inset

 
\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
If we plot the function for intervals with larger and larger 
\begin_inset Formula $x$
\end_inset

 values\SpecialChar \ldots{}
 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/plot_s_0_100.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
\SpecialChar \ldots{}
we see that 
\begin_inset Formula $\ec s$
\end_inset

 oscillates between -1 and 1.
 Still there is no trend to 
\emph on
one
\emph default
 special value.
\end_layout

\begin_layout Standard
It looks as if there is no limit.
\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
Recall the precise definition.
\end_layout

\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Limit at 
\begin_inset Formula $\infty$
\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

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

 if 
\end_layout

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

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec N$
\end_inset

 such that for 
\emph on
every
\emph default
 
\begin_inset Formula $\ec x>N$
\end_inset

 
\end_layout

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

.
 
\end_layout

\end_deeper
\begin_layout Standard
In our case, 
\end_layout

\begin_layout Itemize
For any hypothetical limit 
\begin_inset Formula $\ec L$
\end_inset

, 
\end_layout

\begin_layout Itemize
if 
\begin_inset Formula $\ec\varepsilon=0.9$
\end_inset

, 
\end_layout

\begin_layout Itemize
then regardless of the value of 
\begin_inset Formula $\ec N$
\end_inset

, 
\end_layout

\begin_layout Itemize
we can find infinitely many 
\begin_inset Formula $x$
\end_inset

 values 
\end_layout

\begin_layout Itemize
such that 
\begin_inset Formula $\ec|\sin(x)-L|>\varepsilon$
\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
For any hypothetical limit 
\begin_inset Formula $L$
\end_inset

, if 
\begin_inset Formula $\varepsilon=0.9$
\end_inset

, then regardless of the value of 
\begin_inset Formula $N$
\end_inset

, we can find infinitely many 
\begin_inset Formula $x$
\end_inset

 values such that 
\begin_inset Formula $|\sin(x)-L|>\varepsilon$
\end_inset

.
\end_layout

\begin_layout Standard
For example, if 
\begin_inset Formula $\ec L=0$
\end_inset

, then 
\begin_inset Formula $\ec s\left(\frac{\pi}{2}\right)=1$
\end_inset

, 
\begin_inset Formula $\ec s\left(\frac{5\pi}{2}\right)=1$
\end_inset

, 
\begin_inset Formula $\ec s\left(\frac{9\pi}{2}\right)=1$
\end_inset

, \SpecialChar \ldots{}
.
 This pattern continues for ever, so the limit of the 
\begin_inset Formula $y$
\end_inset

 values does not approach 0 even though 
\begin_inset Formula $\ec y=0$
\end_inset

 infinitely many times at 
\begin_inset Formula $\ec x=0,\pi,2\pi,\ldots$
\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
For any hypothetical limit 
\begin_inset Formula $L$
\end_inset

, if 
\begin_inset Formula $\varepsilon=0.9$
\end_inset

, then regardless of the value of 
\begin_inset Formula $N$
\end_inset

, we can find infinitely many 
\begin_inset Formula $x$
\end_inset

 values such that 
\begin_inset Formula $|\sin(x)-L|>\varepsilon$
\end_inset

.
\end_layout

\begin_layout Standard
We could play this same game with 
\emph on
any
\emph default
 value of 
\begin_inset Formula $\ec L$
\end_inset

.
 (Try it!) So,
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow\infty}s(x)\mathrm{\ does\ not\ exist}}.
\]

\end_inset


\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

Third example
\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
Finally, we'll look at 
\begin_inset Formula $\ec d(x)=1/x$
\end_inset

.
\end_layout

\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
This is an important and useful function.
 Along with its limits, 
\begin_inset Formula $\ec d(x)$
\end_inset

 recurs quite a bit in Calculus.
\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
Use the numerical approach to fill in the question marks, and obtain an
 estimate for 
\begin_inset Formula $\ec\lim_{x\rightarrow\infty}1/x$
\end_inset

.
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
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<features tabularvalignment="middle">
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<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

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</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0 
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

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10.0 
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\end_inset
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
100.0 
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\end_inset
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1000.0 
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\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1,000,000,000.0 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
? 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The values 
\emph on
should
\emph default
 seem to approach a number.
\end_layout

\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 ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
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<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

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</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0 
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\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
100.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1,000,000,000.0 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0 
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</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.1 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.01 
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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.001 
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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0001 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.000000001 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The values 
\emph on
should
\emph default
 seem to approach a number.
\end_layout

\end_deeper
\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_layout Standard
\align center
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="2" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10.0 
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\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
100.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10000.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1,000,000,000.0 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $y$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.0 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.1 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.01 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.001 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0001 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.000000001 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Use the graphical approach to generate two or three plots, and check that
 the guess you'd obtain from the graph agrees with the guess that you'd
 obtain from your table.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
You should have obtained the estimate 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow\infty}\frac{1}{x}=0}.
\]

\end_inset

Let's check this using the precise definition!
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Third example: precise
\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
For any 
\begin_inset Formula $\ec\varepsilon$
\end_inset

, we want to find a value of 
\begin_inset Formula $\ec N$
\end_inset

 such that if 
\begin_inset Formula $\ec x>N$
\end_inset

 then 
\begin_inset Formula $\ec|1/x-0|<\varepsilon$
\end_inset

.
 That is,
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\begin_inset Formula 
\begin{align*}
{\ec\left|\frac{1}{x}\right|} & {\ec<\varepsilon}.
\end{align*}

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\begin_inset Formula 
\begin{align*}
\ec{\dims\left|\frac{1}{x}\right|} & {\dims<\varepsilon}\\
{\ec-\varepsilon<\frac{1}{x}} & {\ec<\varepsilon}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Property of this absolute value inequality.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\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
\noindent
\begin_inset Formula 
\begin{align*}
\ec{\dims-\varepsilon<\frac{1}{x}} & {\dims<\varepsilon}\\
{\ec-\varepsilon x<1} & {\ec<\varepsilon x}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Multiplied both sides by 
\begin_inset Formula $x$
\end_inset

.
\begin_inset Newline newline
\end_inset

 Since 
\begin_inset Formula $x>0$
\end_inset

 as 
\begin_inset Formula $x\rightarrow\infty$
\end_inset

,
\begin_inset Newline newline
\end_inset

 we don't have to change the direction of the inequality.
\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
\noindent
\begin_inset Formula 
\begin{align*}
{\dims-\varepsilon x<1} & {\dims<\varepsilon x}\\
{\ec-\varepsilon x<1} & \mathrm{\ and\ }{\ec1<\varepsilon x}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Meaning of this inequality.
\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
\begin_inset Formula 
\begin{align*}
\ec{\dims-\varepsilon x<1} & \mathrm{\dims\ and\ }{\dims1<\varepsilon x}\\
\mathrm{so\ }{\ec x>-\frac{1}{\varepsilon}} & \mathrm{\ and\ }{\ec x>\frac{1}{\varepsilon}}.
\end{align*}

\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
\noindent
\begin_inset Formula 
\begin{align*}
\ec{\dims-\varepsilon x<1} & {\dims<\varepsilon x}\\
\mathrm{so\ }{\ec x>-\frac{1}{\varepsilon}} & \mathrm{\ and\ }{\ec x>\frac{1}{\varepsilon}}.
\end{align*}

\end_inset

 The larger of these two values is 
\begin_inset Formula $\ec1/\varepsilon$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
If, for example, 
\begin_inset Formula $\ec\varepsilon=0.1$
\end_inset

, then every value of 
\begin_inset Formula $\ec x$
\end_inset

 that is 
\emph on
greater
\emph default
 than 
\begin_inset Formula $\ec1/\varepsilon=10$
\end_inset

 would satisfy 
\begin_inset Formula $\ec|d(x)|<\varepsilon$
\end_inset

.
 (
\emph on
Check
\emph default
 it!) This proves that 
\begin_inset Formula $\ec\lim_{x\rightarrow\infty}d(x)=0$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

BIG-TIME FACT
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
As mentioned, the fact we just proved is something you should remember for
 future reference:
\end_layout

\begin_layout Theorem
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow\infty}\frac{1}{x}=0}.
\]

\end_inset


\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

The other direction
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
In all the examples we looked at in this module, we focused on 
\emph on
positive
\emph default
 infinity.
 Similar arguments can be made for limits at 
\emph on
negative
\emph default
 infinity.
 The definition that you would use is
\end_layout

\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Limit at 
\begin_inset Formula $-\infty$
\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

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

 if 
\end_layout

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

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec N$
\end_inset

 such that for 
\emph on
every
\emph default
 
\begin_inset Formula $\ec x<N$
\end_inset

 
\end_layout

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

.
 
\end_layout

\end_deeper
\begin_layout Standard
\noindent
In this case, multiplying or dividing the inequalities by 
\begin_inset Formula $\ec x$
\end_inset

 would require you to change the direction of the inequality.
\end_layout

\begin_layout BeginFrame
Conclusion
\end_layout

\begin_layout Itemize
When we want to analyze how the 
\begin_inset Formula $y$
\end_inset

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

 behave when 
\begin_inset Formula $\ec x$
\end_inset

 increases (or decreases) without bound, we talk about this as 
\emph on
the limit of 
\begin_inset Formula $\ec f$
\end_inset

 as 
\begin_inset Formula $\ec x$
\end_inset

 approaches infinity (or negative infinity)
\emph default
.
 
\end_layout

\begin_layout Itemize
This limit is defined as
\begin_inset ERT
status open

\begin_layout Plain Layout


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

\end_inset


\begin_inset VSpace -2ex
\end_inset


\end_layout

\begin_deeper
\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Limit at 
\begin_inset Formula $\infty$
\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

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

 if 
\end_layout

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

, 
\end_layout

\begin_layout Itemize
there exists 
\begin_inset Formula $\ec N$
\end_inset

 such that for 
\emph on
every
\emph default
 
\begin_inset Formula $\ec x>N$
\end_inset

 
\end_layout

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

.
 
\end_layout

\end_deeper
\end_deeper
\begin_layout Itemize
As with limits at finite values, there can only be one value for such a
 limit, and they do not always exist.
 
\end_layout

\begin_layout Itemize
All three methods of estimating or evaluating a limit --- numerical, graphical,
 precise --- can be used at infinity.
 
\end_layout

\begin_layout Itemize
An important limit at infinity to remember is 
\begin_inset Formula $\ec\lim_{x\rightarrow\infty}1/x=0$
\end_inset

.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
