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%MODULE 1: Epsilon-Delta limits
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\begin_layout Title
An intuitive notion of limits
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\begin_layout Subtitle
In this lesson we introduce the 
\emph on
fundamental
\emph default
 concept of Calculus: 
\emph on

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 the limit
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.
 For now we will take a more intuitive approach, looking at 
\emph on
graphical
\emph default
 and 
\emph on
numerical
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 approaches to the limit.
 We will introduce a rigorous, 
\emph on
algebraic
\emph default
 approach to the limit in another module.
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Two sample problems
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\noindent
Let 
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\begin_layout Itemize
\begin_inset Formula $\ec{\displaystyle {\ec f\left(x\right)=\frac{x^{2}-x}{x}}}$
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, and 
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\begin_inset Formula $\ec{\displaystyle {\ec w\left(x\right)=\sin\left(\frac{1}{x}\right)}}$
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.
 
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Both of these functions are undefined 
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at
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\begin_inset Formula $\ec x=0$
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.
 We want to explore how they behave as 
\begin_inset Formula $\ec x$
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approaches
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\begin_inset Formula $\ec0$
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.
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The limit
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[
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(Intuitive definition of a limit)
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The 
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 limit
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}
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 indicates the value to which a variable or a function grows arbitrarily
 close.
 We write 
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=L$
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 to mean that the 
\begin_inset Formula $y$
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 values of 
\begin_inset Formula $\ec f(x)$
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 grow arbitrarily close to 
\begin_inset Formula $\ec L$
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 when the 
\begin_inset Formula $x$
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 values grow arbitrarily close to 
\begin_inset Formula $\ec a$
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.
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\noindent
In the case examples listed on the previous page, we want to know the 
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limit
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\begin_inset Formula $\ec f$
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 or 
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 as 
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In general, when we want to know the 
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 as 
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 approaches some value 
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, we write
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\[
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\]

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In this worksheet, we want to evaluate
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\[
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\]

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SAGE worksheets
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\noindent
In this lab you will not need a special SAGE worksheet, but you will need
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 open a new worksheet.
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The numerical approach
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\noindent
The first method to evaluating a limit is called the 
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 To estimate the limit of 
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 as 
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 approaches 
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-
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 values using values of 
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 that get close to 
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 We should choose values that lie on both sides of 
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, in order to make sure the limits are the same from left 
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?
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It 
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\begin_inset Formula $y$
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\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
You can find these values by hand.
 This function isn't very difficult, so that's easy.
 
\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 Itemize
You can use a calculator.
 This can be rather tiresome on the fingers when the function is involved,
 but it probably saves times from hand computation.
 (If your calculator has a table feature, your fingers won't be so tired.)
 
\end_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 Itemize
You can also use SAGE.
 How? 
\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 Itemize
You can also use SAGE.
 How? 
\end_layout

\begin_deeper
\begin_layout Itemize
The simplest way is to define the function 
\begin_inset Formula $\ec f$
\end_inset

 by typing 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 f = (x‸2-x)/x
\family default
.
 To evaluate it at 
\begin_inset Formula $\ec x=-1$
\end_inset

, type 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 f(-1)
\family default
.
 Repeat this for other values.
 
\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 Itemize
You can also use SAGE.
 How? 
\end_layout

\begin_deeper
\begin_layout Itemize
The simplest way is to define the function 
\begin_inset Formula $\ec f$
\end_inset

 by typing 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 f = (x‸2-x)/x
\family default
.
 To evaluate it at 
\begin_inset Formula $\ec x=-1$
\end_inset

, type 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 f(-1)
\family default
.
 Repeat this for other values.
 
\end_layout

\begin_layout Itemize
A somewhat faster method is to use a 
\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

 loop
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 You can do this by typing 
\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,-0.1,-0.01,0.01,0.1,1]]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 (
\emph on
Try it!
\emph default
) In this 
\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

 for
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 loop, the variable 
\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

 a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 takes all the values specified in the list, and the expression 
\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)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 is evaluated for each of those values.
 
\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

The graphical approach
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Now let's apply the graphical approach to 
\begin_inset Formula $\ec f(x)$
\end_inset

.
 Sketch a graph of 
\begin_inset Formula $\ec f$
\end_inset

 around 
\begin_inset Formula $\ec x=0$
\end_inset

.
 You should see something like this: 
\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.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
The graphical approach: WARNING
\end_layout

\begin_layout Standard
\noindent
\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

 Now let's apply the graphical approach to 
\begin_inset Formula $f(x)$
\end_inset

.
 Sketch a graph of 
\begin_inset Formula $f$
\end_inset

 around 
\begin_inset Formula $x=0$
\end_inset

.
 You should see something like this:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\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.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
You have to be careful reading plots that are generated by a computer.
 The computer doesn't know how to discover the division by zero, so it draws
 the curve as if there were a point at 
\begin_inset Formula $\ec(0,-1)$
\end_inset

 
\emph on
even though there is no point there
\emph default
.
\end_layout

\begin_layout BeginFrame
The graphical approach: the limit
\end_layout

\begin_layout Standard
\noindent
\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

 Now let's apply the graphical approach to 
\begin_inset Formula $f(x)$
\end_inset

.
 Sketch a graph of 
\begin_inset Formula $f$
\end_inset

 around 
\begin_inset Formula $x=0$
\end_inset

.
 You should see something like this:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\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_twosided.pdf
	lyxscale 20
	scale 6.25

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
It is often easy to 
\begin_inset Quotes eld
\end_inset

see
\begin_inset Quotes erd
\end_inset

 the limit from a graph.
 Recall that 
\begin_inset Formula $\ec\lim_{x\rightarrow0}f(x)$
\end_inset

 is the value that 
\begin_inset Formula $\ec f$
\end_inset

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

 approaches 
\begin_inset Formula $\ec0$
\end_inset

.
 Imagine that you are an ant walking along the graph of 
\begin_inset Formula $\ec f$
\end_inset

.
 What 
\begin_inset Formula $y$
\end_inset

 value would you approach when your 
\begin_inset Formula $x$
\end_inset

 value approaches 
\begin_inset Formula $0$
\end_inset

? That point is the 
\emph on
limit
\emph default
 of 
\begin_inset Formula $\ec f$
\end_inset

 as 
\begin_inset Formula $\ec x$
\end_inset

 approaches 
\begin_inset Formula $0$
\end_inset

.
 Here 
\begin_inset Formula 
\[
\ec{\lim_{x\rightarrow0}f(x)=-1}.
\]

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

SAGE and plots
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
As with the numerical approach, you have several options for sketching a
 graph.
 
\end_layout

\begin_layout Itemize
You can sketch it by hand using an 
\begin_inset Formula $x$
\end_inset

-
\begin_inset Formula $y$
\end_inset

 table.
 This takes a long time, and is subject to human error.
 
\end_layout

\begin_layout Itemize
You can use a graphing calculator.
 This doesn't take very long, but requires a $100
\begin_inset space ~
\end_inset

calculator that isn't very flexible in its graphing.
 
\end_layout

\begin_layout Itemize
You can use SAGE's 
\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

 plot()
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 function.
 This doesn't take very long, does not require any money besides what you've
 already paid for the class, and is very flexible.
 See the module 
\begin_inset Quotes eld
\end_inset

An Introduction to SAGE
\begin_inset Quotes erd
\end_inset

 for more details.
 
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Specificity of a limit
\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
Notice that we have talked about computing or looking from 
\emph on
both
\emph default
 sides.
 The limit can have only 
\emph on
one specific
\emph default
 value.
 It should be the same whether 
\begin_inset Formula $\ec x$
\end_inset

 approaches 
\begin_inset Formula $\ec a$
\end_inset

 from negative values (
\begin_inset Quotes eld
\end_inset

from the left
\begin_inset Quotes erd
\end_inset

) or whether 
\begin_inset Formula $\ec x$
\end_inset

 approaches 
\begin_inset Formula $\ec a$
\end_inset

 from positive values (
\begin_inset Quotes eld
\end_inset

from the right
\begin_inset Quotes erd
\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
\noindent
We write this property as 
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)
\]

\end_inset

 where 
\begin_inset Formula $\ec x\rightarrow a^{-}$
\end_inset

 reads 
\begin_inset Quotes eld
\end_inset

as 
\begin_inset Formula $\ec x$
\end_inset

 approaches 
\begin_inset Formula $\ec a$
\end_inset

 from the left
\begin_inset Quotes erd
\end_inset

 and 
\begin_inset Formula $\ec x\rightarrow a^{+}$
\end_inset

 reads 
\begin_inset Quotes eld
\end_inset

as 
\begin_inset Formula $\ec x$
\end_inset

 approaches 
\begin_inset Formula $\ec a$
\end_inset

 from the right
\begin_inset Quotes erd
\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
Both the table of 
\begin_inset Formula $x$
\end_inset

-
\begin_inset Formula $y$
\end_inset

 values and the graph suggested that 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0^{-}}f(x)=\lim_{x\rightarrow0^{+}}f(x)},
\]

\end_inset

 so we were able to conclude that 
\begin_inset Formula $\ec{\displaystyle \lim_{x\rightarrow0}f(x)=0}$
\end_inset

; that is, the 
\emph on
two-sided limit
\emph default
 exists.
\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
On the other hand, the function 
\begin_inset Formula $\ec g$
\end_inset

 graphed below has 
\emph on
no
\emph default
 limit at 
\begin_inset Formula $\ec x=0$
\end_inset

, because the limits from the left and right disagree.
 
\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/twosided_necessary.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
 
\begin_inset Formula ${\displaystyle \lim_{x\rightarrow0^{-}}g(x)\neq\lim_{x\rightarrow0^{+}}g(x)}$
\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\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 Standard
\noindent
Now let's try to estimate
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow0}w(x)
\]

\end_inset

where
\begin_inset Formula 
\[
{\ec w(x)=\sin\left(\frac{1}{x}\right)}.
\]

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Second example: The numerical 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
\noindent
To estimate the limit of 
\begin_inset Formula $\ec w$
\end_inset

 as 
\begin_inset Formula $\ec x$
\end_inset

 approaches 
\begin_inset Formula $0$
\end_inset

, we build a table of 
\begin_inset Formula $x$
\end_inset

-
\begin_inset Formula $y$
\end_inset

 values using values of 
\begin_inset Formula $\ec x$
\end_inset

 that get close to 
\begin_inset Formula $\ec0$
\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
Fill in the table, rounding to the ten-thousandths place.
\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="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" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1.0 
\end_layout

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

\begin_layout Plain Layout
-0.1 
\end_layout

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

\begin_layout Plain Layout
-0.01 
\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

 0
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

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

\begin_layout Plain Layout
0.01 
\end_layout

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

\begin_layout Plain Layout
0.1 
\end_layout

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

\begin_layout Plain Layout
1.0 
\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" 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>
<cell alignment="center" valignment="top" topline="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
\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 ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

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

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" 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


\begin_inset Newline newline
\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
Fill in the table, rounding to the ten-thousandths place.
\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="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" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1.0 
\end_layout

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

\begin_layout Plain Layout
-0.1 
\end_layout

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

\begin_layout Plain Layout
-0.01 
\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

 0
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

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

\begin_layout Plain Layout
0.01 
\end_layout

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

\begin_layout Plain Layout
0.1 
\end_layout

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

\begin_layout Plain Layout
1.0 
\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" usebox="none">
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-0.8415 
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0.5540 
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0.5063 
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{
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\begin_inset ERT
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\backslash
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\end_inset

 ???
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}
\end_layout

\end_inset

 
\end_layout

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-0.5440 
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0.8415 
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset Newline newline
\end_inset

 Can you guess the value of 
\begin_inset Formula $\ec\lim_{x\rightarrow0}w(x)$
\end_inset

? 
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Fill in the table, rounding to the ten-thousandths place.
\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
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\end_inset


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-1.0 
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 0
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}
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0.01 
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0.1 
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1.0 
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-0.5540 
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-0.5063 
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{
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\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 0.5?
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status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

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0.5063 
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\begin_layout Plain Layout
0.5440 
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0.8415 
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\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
You might try to 
\begin_inset Quotes eld
\end_inset

guess
\begin_inset Quotes erd
\end_inset

 from this data that 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0}w(x)\approx0.5}.
\]

\end_inset

You might also try to 
\begin_inset Quotes eld
\end_inset

guess
\begin_inset Quotes erd
\end_inset

 that it should be the average; that is, 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0}w(x)=0}.
\]

\end_inset

 
\emph on

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 That would be a very bad guess.
\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

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

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
You should 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_w.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
What is going on?
\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

The infinite wiggle
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
This graph is called an 
\emph on
infinite wiggle
\emph default
.
 As the curve gets closer to 
\begin_inset Formula $\ec x=0$
\end_inset

, the 
\begin_inset Formula $y$
\end_inset

-values start to vibrate more and more wildly between 
\begin_inset Formula $\ec y=-1$
\end_inset

 and 
\begin_inset Formula $\ec y=1$
\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_w.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
The infinite wiggle zoomed
\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
This graph is something called an 
\emph on
infinite wiggle
\emph default
.
 As the curve gets closer to 
\begin_inset Formula $\ec x=0$
\end_inset

, the 
\begin_inset Formula $y$
\end_inset

-values start to vibrate more and more wildly between 
\begin_inset Formula $\ec y=-1$
\end_inset

 and 
\begin_inset Formula $\ec y=1$
\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
\noindent
\begin_inset Float figure
placement h!
wide false
sideways false
status open

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

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
You can zoom in closer if you wish; 
\begin_inset Formula $\ec w$
\end_inset

 wiggles all the more.
 It's a regular party down there.
\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
\noindent
\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_w.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
What then should we say about the limit of 
\begin_inset Formula $\ec w$
\end_inset

 as 
\begin_inset Formula $\ec x$
\end_inset

 approaches 
\begin_inset Formula $0$
\end_inset

? The function does not approach 
\emph on
one
\emph default
 specific value, so we say that 
\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 limit 
\emph on
does not exist
\emph default

\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

Fixing the numerical approach
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
We can see this clearly by computing more 
\begin_inset Formula $x$
\end_inset

-
\begin_inset Formula $y$
\end_inset

 values in the numerical approach.
 Here is a slightly larger table of values approaching from the left:
\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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-1.0 
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-0.1 
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-0.01 
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-0.00001 
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\end_inset

 
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\begin_inset Formula $0.5064$
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

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

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Use both methods carefully
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
To estimate a limit, it is probably a good idea to use both methods.
 You should build 
\begin_inset Formula $x$
\end_inset

-
\begin_inset Formula $y$
\end_inset

 tables with a 
\emph on
lot
\emph default
 of carefully chosen 
\begin_inset Formula $\ec x$
\end_inset

-values, and you should zoom into your graphs to make sure that no strange
 behaviors are disguised by too large a viewing area.
 
\end_layout

\begin_layout Itemize
Sometimes numerical values can appear to bounce around, but a graph will
 show that there is a trend for the bouncing to even out as the 
\begin_inset Formula $y$
\end_inset

-values approach a definite number.
 For an example of this phenomenon, look at a plot of the function 
\begin_inset Formula $\ec x(1+\sin(100x))$
\end_inset

 on the interval 
\begin_inset Formula $\ec[-1,1]$
\end_inset

.
 
\end_layout

\begin_layout Itemize
On the other hand, the graph of a function may sometimes suggest a limit
 when none exists, whereas a numerical approach will show that the values
 are in fact bouncing around.
 For an example of this phenomenon, look at a plot of the function 
\begin_inset Formula $\ec0.01\sin(1/x)+x$
\end_inset

 on the interval 
\begin_inset Formula $\ec[-1,1]$
\end_inset

.
 
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Estimation vs.
 evaluation
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
All the same, these methods only allow us to 
\emph on
estimate
\emph default
 a limit, they do not allow us to 
\emph on
evaluate
\emph default
 the limit.
 We will develop a precise method to help us evaluate limits 
\emph on
precisely
\emph default
 in a future module.
\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
The 
\emph on
limit
\emph default
 of a function 
\begin_inset Formula $\ec f$
\end_inset

 as 
\begin_inset Formula $\ec x$
\end_inset

 approaches a value 
\begin_inset Formula $\ec a$
\end_inset

 is the value 
\begin_inset Formula $\ec f$
\end_inset

 approaches as we close in on 
\begin_inset Formula $\ec x=a$
\end_inset

 from the left and right sides of 
\begin_inset Formula $\ec a$
\end_inset

.
 
\end_layout

\begin_layout Itemize
There can only be one value for a limit.
 Limits do not always exist.
 
\end_layout

\begin_layout Itemize
We studied two methods of 
\emph on
estimating
\emph default
 limits: 
\end_layout

\begin_deeper
\begin_layout Itemize
a 
\emph on
numerical
\emph default
 method, and 
\end_layout

\begin_layout Itemize
a 
\emph on
graphical
\emph default
 method.
 
\end_layout

\end_deeper
\begin_layout Itemize
Using both these methods can give us an idea of what the value of a limit
 is, but they do not give us 
\emph on
certainty
\emph default
 as to what the value of a limit is.
 
\end_layout

\begin_layout EndFrame

\end_layout

\end_body
\end_document
