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%MODULE: Continuity
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\begin_layout Title
Properties of the Limit
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\begin_layout Subtitle
We use limits a great deal in Calculus.
 As a result, we need to analyze the limit carefully and identify any properties
 that might prove useful.
 We will in fact apply some of the properties discussed here in future modules.
 
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Defining the problem
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\noindent
We would like to evaluate the following limits 
\emph on
without
\emph default
 numerical or graphical approximation, and 
\emph on
without
\emph default
 verifying 
\begin_inset Formula $\ec\epsilon$
\end_inset

 and 
\begin_inset Formula $\ec\delta$
\end_inset

.
 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\lim_{x\rightarrow0}c$
\end_inset

, where 
\begin_inset Formula $\ec c$
\end_inset

 is any real number.
 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\lim_{h\rightarrow0}\left(e^{x}f(h)\right)$
\end_inset

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

 is any real number and 
\begin_inset Formula $\ec f$
\end_inset

 is any function in terms of 
\begin_inset Formula $\ec h$
\end_inset

 
\emph on
and not 
\begin_inset Formula $\ec x$
\end_inset


\emph default
.
 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\lim_{h\rightarrow0}\frac{f(x)}{g(x)}$
\end_inset

 when we know 
\begin_inset Formula $\ec\lim_{h\rightarrow0}f(x)$
\end_inset

 and 
\begin_inset Formula $\ec\lim_{h\rightarrow0}g(x)$
\end_inset

.
 
\end_layout

\begin_layout Itemize
Other algebraic combinations of 
\begin_inset Formula $\ec f$
\end_inset

 and 
\begin_inset Formula $\ec g$
\end_inset

.
 
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SAGE worksheets
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\begin_layout Standard
\noindent
You should open the SAGElet 
\begin_inset Quotes eld
\end_inset

Limits: 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset


\begin_inset Quotes erd
\end_inset

.
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{
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A word on continuity
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}
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\noindent
If you have not studied the module on 
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 continuity
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 yet, you can ignore this slide.
 Otherwise, read on.
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You instructor may have chosen to discuss with you the topic of 
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 continuity
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}
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 before covering this material.
 If so, you may notice that some of the properties we will give seem redundant
 with what was discussed in that slide.
 This is not really true.
 We gave in that module a theorem that certain functions were continuous.
 We did not prove the theorem, however, and with good reason: the proof
 of that theorem relies on properties of the limit described below.
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A word on proofs
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\begin_layout Standard
\noindent
If you have not studied the module on 
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{
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 limits using 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset


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}
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\end_inset

, you can ignore any slide that talks about 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset

.
 Many instructors choose not to dwell on this approach.
\end_layout

\begin_layout Standard
It is possible to prove all of the properties that we describe in this module
 using an 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset

 argument.
 We will prove only a few of them this way; the others we will state without
 any argument at all.
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First properties
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\begin_deeper
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\noindent
The first properties that we describe deal with a 
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 constant
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 or a 
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 constant multiple
\begin_inset ERT
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}
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\end_inset

.
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\begin_layout Theorem
Let 
\begin_inset Formula $\ec c$
\end_inset

 be a constant with respect to 
\begin_inset Formula $\ec x$
\end_inset

.
 Then for any 
\begin_inset Formula $\ec a\in\mathbb{R}$
\end_inset

 and for any function 
\begin_inset Formula $\ec f(x)$
\end_inset

, 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec\lim_{x\rightarrow a}c=c$
\end_inset

; and 
\end_layout

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

, if the latter limit exists.
 
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<2>
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\begin_deeper
\begin_layout Standard
For an example of the first property, 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow3}4=4}.
\]

\end_inset


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\begin_deeper
\begin_layout Standard
For an example of the second property,
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow1}4x^{2}=4\lim_{x\rightarrow1}x^{2}=4\cdot1=4}.
\]

\end_inset


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\begin_deeper
\begin_layout Standard
Another example of the second property that we will use in a future module
 is
\begin_inset Formula 
\[
{\ec\lim_{h\rightarrow0}\left(e^{x}f(h)\right)=e^{x}\lim_{h\rightarrow0}f(h)}
\]

\end_inset

 where 
\begin_inset Formula $\ec f$
\end_inset

 is a function in terms of 
\begin_inset Formula $\ec h$
\end_inset

 
\emph on
and not 
\begin_inset Formula $\ec x$
\end_inset


\emph default
.
 Notice that 
\begin_inset Formula $\ec e^{x}$
\end_inset

 depends only on 
\begin_inset Formula $\ec h$
\end_inset

, so it is constant 
\emph on
with respect to 
\begin_inset Formula $\ec h$
\end_inset


\emph default
.
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Proof of the limit of a constant: concept
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\begin_deeper
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\noindent
It is relatively easy to prove these two properties using an 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset

 argument.
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<1>
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\begin_deeper
\begin_layout Standard
We remind you that the 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset

 argument comes from the precise definition of a limit:
\end_layout

\begin_layout Definition
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[
\end_layout

\end_inset

(Precise definition of the limit)
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]
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\end_inset


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

 if 
\end_layout

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

, 
\end_layout

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

 
\end_layout

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

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

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

), 
\end_layout

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

.
 
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<2->
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\begin_deeper
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\noindent
In the SAGE worksheet 
\begin_inset Quotes eld
\end_inset

Limits: 
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset


\begin_inset Quotes erd
\end_inset

, set 
\begin_inset Formula $\ec f$
\end_inset

 to be the constant 
\begin_inset Formula $\ec1$
\end_inset

.
 (To do this, you must type 
\family typewriter

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\backslash
ttcolor
\end_layout

\end_inset

 lambda x: 1
\family default
 in the 
\family typewriter

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\backslash
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\end_layout

\end_inset

 f
\family default
 box.
 This strange notation is due to a bug which ought to be resolved soon.)
 You should see the following plot.
 
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\begin_layout Plain Layout
 
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	filename figures/c.png
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	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

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\begin_layout Pause

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<3>
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\end_layout

\begin_deeper
\begin_layout Standard
Look carefully at the plot.
 Notice that the graph of 
\begin_inset Formula $\ec f$
\end_inset

 does not pass through the pink regions.
 This means that you have found an appropriate value of 
\begin_inset Formula $\ec\delta$
\end_inset

 to guarantee that the distance between 
\begin_inset Formula $\ec L$
\end_inset

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

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

, satisfying the definition of the limit.
\end_layout

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\end_layout

\begin_deeper
\begin_layout Standard
Now look more carefully.
 Experiment with different values of 
\begin_inset Formula $\ec\delta$
\end_inset

.
 Can you find any value of 
\begin_inset Formula $\ec\delta$
\end_inset

 such that 
\begin_inset Formula $\ec f(x)$
\end_inset

 passes through the pink regions?
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
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<5>
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\end_layout

\begin_deeper
\begin_layout Standard
You should not be able to find such a value.
 In fact, 
\begin_inset Formula $\ec|f(x)-L|=|1-1|=0<\epsilon$
\end_inset

 
\emph on
regardless of the value of 
\begin_inset Formula $\ec\epsilon$
\end_inset


\emph default
.
\end_layout

\end_deeper
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Proof of the limit of a constant
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\begin_layout Standard
\noindent
We can generalize this argument to show that for any constant function 
\begin_inset Formula $\ec f(x)=c$
\end_inset

 and for any real number 
\begin_inset Formula $\ec a$
\end_inset

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

.
 Let 
\begin_inset Formula $\ec\epsilon$
\end_inset

 be any positive real number, and then
\begin_inset Formula 
\[
{\ec|f(x)-L|=|c-c|=0<\epsilon}.
\]

\end_inset

 You can choose 
\emph on
any
\emph default
 value of 
\begin_inset Formula $\ec\delta$
\end_inset

 and satisfy the precise definition of the limit.
\end_layout

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Proof of the limit of a constant multiple: concept
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\begin_deeper
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\noindent
To determine how to prove the limit of a constant multiple, we'll take a
 similar approach, and try an example function and an example constant.
 Let 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

, 
\begin_inset Formula $\ec a=1$
\end_inset

, and 
\begin_inset Formula $\ec c=-2$
\end_inset

.
 We want to see if we can show that 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow1}(-2x^{2})=-2\lim_{x\rightarrow1}x^{2}}.
\]

\end_inset


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<2-3>
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\begin_deeper
\begin_layout Standard
A glance at the graph suggests that 
\begin_inset Formula $\ec\lim_{x\rightarrow1}x^{2}=1$
\end_inset

.
 Look again at the SAGElet, and start with 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

.
 Using 
\begin_inset Formula $\ec\epsilon=0.1$
\end_inset

, find 
\begin_inset Formula $\ec\delta$
\end_inset

 satisfying the definition of the limit.
\end_layout

\begin_layout Pause

\end_layout

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<3>
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	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset VSpace -3ex
\end_inset

You should find that 
\begin_inset Formula $\ec\delta\approx0.05$
\end_inset

 works.
 (In fact the correct answer is 
\begin_inset Formula $\ec\delta=-1+\sqrt{1.1}\approx0.0488$
\end_inset

.)
\end_layout

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\begin_deeper
\begin_layout Standard
Now try the SAGElet with 
\begin_inset Formula $\ec cf(x)=-2x^{2}$
\end_inset

.
 Keep 
\begin_inset Formula $\ec\delta=0.05$
\end_inset

.
 What value of 
\begin_inset Formula $\ec\epsilon$
\end_inset

 works with this 
\begin_inset Formula $\ec\delta$
\end_inset

 to guarantees that 
\begin_inset Formula $\ec|cf(x)-cL|<\epsilon$
\end_inset

? (Note 
\begin_inset Formula $\ec cL=-2\cdot1=-2$
\end_inset

.)
\end_layout

\begin_layout Pause

\end_layout

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\begin_inset ERT
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<5>
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\end_inset


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\begin_deeper
\begin_layout Standard
\begin_inset VSpace -2ex
\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
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	filename figures/neg2x2_good.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset VSpace -3ex
\end_inset

You should find that 
\begin_inset Formula $\ec\epsilon=0.2$
\end_inset

.
 This is twice the value of the previous epsilon.
\end_layout

\end_deeper
\end_deeper
\begin_layout Pause

\end_layout

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\begin_inset ERT
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{
\end_layout

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\backslash
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\end_layout

\end_inset

 You should find that 
\begin_inset Formula $\epsilon=0.2$
\end_inset

.
 This is twice the value of the previous epsilon.
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Perhaps the 
\begin_inset Formula $\ec\delta$
\end_inset

 that makes 
\begin_inset Formula $\ec\epsilon$
\end_inset

 work for 
\begin_inset Formula $\ec cf(x)$
\end_inset

, is the same as the 
\begin_inset Formula $\ec\delta$
\end_inset

 that makes 
\begin_inset Formula $\ec\epsilon/|c|$
\end_inset

 work for 
\begin_inset Formula $\ec f(x)$
\end_inset

.
\end_layout

\end_deeper
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Proof of the limit of a constant multiple
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\begin_deeper
\begin_layout Standard
\noindent
We try this in a proof.
 We want to show that 
\begin_inset Formula $\ec\lim_{x\rightarrow a}cf(x)=c\lim_{x\rightarrow a}f(x)$
\end_inset

, if the latter limit exists.
 Assume that the latter limit 
\emph on
does
\emph default
 exist; that is, 
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)=L$
\end_inset

.
\end_layout

\begin_layout Only
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<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We need to show that 
\begin_inset Formula $\ec\lim_{x\rightarrow a}cf(x)=cL$
\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
If 
\begin_inset Formula $\ec c=0$
\end_inset

 then 
\begin_inset Formula $\ec cf(x)=0$
\end_inset

, and we are talking about the limit of a constant.
 We have already discussed this case, so we can assume that 
\begin_inset Formula $\ec c\neq0$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
By the definition of a limit, we must show that for every positive 
\begin_inset Formula $\ec\epsilon$
\end_inset

 we can find a positive 
\begin_inset Formula $\ec\delta$
\end_inset

 satisfying
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\[
{\ec|x-a|<\delta}\qquad{\ec\Longrightarrow}\qquad{\ec|cf(x)-cL|<\epsilon}.
\]

\end_inset


\begin_inset VSpace -1ex
\end_inset

Let 
\begin_inset Formula $\ec\epsilon'=\epsilon/|c|$
\end_inset

; this is a positive number.
 Since 
\begin_inset Formula $\ec L$
\end_inset

 is the limit of 
\begin_inset Formula $\ec f(x)$
\end_inset

 as 
\begin_inset Formula $\ec x\rightarrow a$
\end_inset

, the precise definition tells us that there exists 
\begin_inset Formula $\ec\delta$
\end_inset

 such that
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\[
{\ec|x-a|<\delta}\qquad{\ec\Longrightarrow}\qquad{\ec|f(x)-L|<\epsilon'}.
\]

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset VSpace -1ex
\end_inset

Then
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\[
{\ec|cf(x)-cL|=|c(f(x)-L)|=|c|\cdot|f(x)-L|<|c|\cdot\epsilon'=\epsilon}.
\]

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset VSpace -1ex
\end_inset

We have found 
\begin_inset Formula $\ec\delta$
\end_inset

 such that if 
\begin_inset Formula $\ec|x-a|<\delta$
\end_inset

, then 
\begin_inset Formula $\ec|cf(x)-cL|<\epsilon$
\end_inset

.
 We have proved the second part of the theorem:
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow a}cf(x)=cL=c\lim_{x\rightarrow a}f(x)}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

The limit of 
\begin_inset Formula $x$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Our next property is very convenient.
\end_layout

\begin_layout Theorem
\noindent
For all 
\begin_inset Formula $\ec a\in\mathbb{R}$
\end_inset

,
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow a}x=a}.
\]

\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
The graph of 
\begin_inset Formula $\ec f(x)=x$
\end_inset

 gives credence to the theorem.
 
\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/x_around_0.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
 If you approach 
\begin_inset Formula $\ec x=0$
\end_inset

 from both sides, it looks as if the limit is 
\begin_inset Formula $\ec0$
\end_inset

.
 
\end_layout

\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

<2->
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Let's experiment a little with the SAGElet to see how 
\begin_inset Formula $\ec\delta$
\end_inset

 corresponds to 
\begin_inset Formula $\ec\epsilon$
\end_inset

.
 Change 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 f
\family default
 to 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 x
\family default
, and pick any 
\begin_inset Formula $x$
\end_inset

-value you like.
\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
What is the largest value of 
\begin_inset Formula $\ec\delta$
\end_inset

 that satisfies the precise definition of the limit for 
\begin_inset Formula $\ec\epsilon=\hilite0.1$
\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
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/x_epsilon1.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
It looks here as if 
\begin_inset Formula $\ec\delta=0.1$
\end_inset

 works.
 What if we change 
\begin_inset Formula $\ec\epsilon$
\end_inset

 to 
\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

 0.01
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\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

<5-6>
\end_layout

\end_inset


\end_layout

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

 that satisfies the precise definition of the limit for 
\begin_inset Formula $\epsilon={\hilite0.01}$
\end_inset

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/x_epsilon2.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\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
It looks here as if 
\begin_inset Formula $\ec\delta=0.01$
\end_inset

 works.
 What if we change 
\begin_inset Formula $\ec\epsilon$
\end_inset

 to 
\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

 0.5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\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

<7-8>
\end_layout

\end_inset


\end_layout

\begin_layout Standard
What is the value of 
\begin_inset Formula $\delta$
\end_inset

 that satisfies the precise definition of the limit for 
\begin_inset Formula $\epsilon={\hilite0.5}$
\end_inset

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/x_epsilon3.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\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
It looks here as if 
\begin_inset Formula $\ec\delta=0.5$
\end_inset

 works.
\end_layout

\end_deeper
\end_deeper
\end_deeper
\begin_layout BeginFrame
The limit of 
\begin_inset Formula $x$
\end_inset


\end_layout

\begin_layout Standard
\noindent
Our next property is very convenient.
\end_layout

\begin_layout Theorem
\noindent
For all 
\begin_inset Formula $a\in\mathbb{R}$
\end_inset

,
\begin_inset Formula 
\[
{\lim_{x\rightarrow a}x=a}.
\]

\end_inset

 
\end_layout

\begin_layout Standard
In each case, 
\begin_inset Formula $\ec\delta$
\end_inset

 had the same value as 
\begin_inset Formula $\ec\epsilon$
\end_inset

.
 Let's try that in a proof.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Let 
\begin_inset Formula $\ec\epsilon$
\end_inset

 be any positive real number, and set 
\begin_inset Formula $\ec\delta=\epsilon$
\end_inset

.
 Assume that 
\begin_inset Formula $\ec|x-a|<\epsilon$
\end_inset

.
 Then
\begin_inset Formula 
\[
{\ec|f(x)-L|=|x-a|}.
\]

\end_inset

 By hypothesis, 
\begin_inset Formula $\ec|x-a|<\delta=\epsilon$
\end_inset

, so
\begin_inset Formula 
\[
{\ec|f(x)-L|<\epsilon}.
\]

\end_inset

 We have thus proved the theorem.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

More properties
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Here are most remaining properties.
 We do not prove them.
\end_layout

\begin_layout Theorem
\noindent
Let 
\begin_inset Formula $\ec a\in\mathbb{R}$
\end_inset

, and 
\begin_inset Formula $\ec f$
\end_inset

 and 
\begin_inset Formula $\ec g$
\end_inset

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

 and 
\begin_inset Formula $\ec\lim_{x\rightarrow a}g(x)=M$
\end_inset

.
 
\end_layout

\begin_deeper
\begin_layout Itemize
The limit of a sum is the sum of the limits:
\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula $\ec\lim_{x\rightarrow a}\left(f(x)\pm g(x)\right)=L+M$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Itemize
\begin_inset VSpace -1ex
\end_inset

The limit of a product is the product of the limits:
\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula $\ec\lim_{x\rightarrow a}\left(f(x)g(x)\right)=LM$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Itemize
\begin_inset VSpace -1ex
\end_inset

The limit of a quotient is the quotient of the limits:
\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset VSpace -1ex
\end_inset

If 
\begin_inset Formula $\ec M\neq0$
\end_inset

, then 
\begin_inset Formula $\ec\lim_{x\rightarrow a}\left(f(x)/g(x)\right)=L/M$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Itemize
\begin_inset VSpace -1ex
\end_inset

If 
\begin_inset Formula $\ec n\in\mathbb{N}$
\end_inset

 then 
\begin_inset Formula $\ec\lim_{x\rightarrow a}x^{n}=a^{n}$
\end_inset

 and 
\begin_inset Formula $\ec\lim_{x\rightarrow a}\sqrt[n]{x}=\sqrt[n]{L}$
\end_inset

.
 
\end_layout

\begin_layout Itemize
\begin_inset VSpace -1ex
\end_inset

If 
\begin_inset Formula $\ec n\in\mathbb{N}$
\end_inset

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

 and 
\begin_inset Formula $\ec\lim_{x\rightarrow a}\sqrt[n]{f(x)}=\sqrt[n]{L}$
\end_inset

,
\begin_inset Newline newline
\end_inset

 
\emph on
provided that
\emph default
 
\begin_inset Formula $\ec\sqrt[n]{L}$
\end_inset

 exists.
 
\end_layout

\begin_layout Itemize
\begin_inset VSpace -1ex
\end_inset

If 
\begin_inset Formula $\ec f$
\end_inset

 is a 
\emph on
polynomial
\emph default
 or 
\emph on
rational expression
\emph default
 and 
\begin_inset Formula $\ec a$
\end_inset

 is in the domain of 
\begin_inset Formula $\ec f$
\end_inset

,
\begin_inset Newline newline
\end_inset

 then you can substitute the value 
\begin_inset Formula $\ec x=a$
\end_inset

 into 
\begin_inset Formula $\ec f$
\end_inset

 to find the limit:
\end_layout

\begin_deeper
\begin_layout Standard
\align center
\begin_inset VSpace -1ex
\end_inset


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

.
\end_layout

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

\begin_layout Plain Layout

{
\end_layout

\end_inset

Examples
\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 Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
Suppose we have two probes whose value at 
\begin_inset Formula $\ec x$
\end_inset

 is modeled by the functions 
\begin_inset Formula $\ec f(x)$
\end_inset

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

, and we know that 
\begin_inset Formula $\ec\lim_{x\rightarrow3}f(x)=-2.5$
\end_inset

 and 
\begin_inset Formula $\ec\lim_{x\rightarrow3}g(x)=0$
\end_inset

.
 Then
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\begin{align*}
\ec\lim_{x\rightarrow3}\left(f(x)+g(x)\right) & \ec=-2.5\\
\ec\lim_{x\rightarrow3}\frac{g(x)}{f(x)} & \ec=0\\
\ec\lim_{x\rightarrow3}\sqrt[4]{-f(x)} & {\ec=\sqrt[4]{-\lim_{x\rightarrow3}f(x)}\approx1.2574}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

\end_layout

\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 have to be careful with division by zero.
 In particular, we 
\emph on
cannot
\emph default
 say that
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\[
\ec\lim_{x\rightarrow3}\frac{f(x)}{g(x)}
\]

\end_inset


\begin_inset VSpace -1ex
\end_inset

does not exist.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3-6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Limits 
\emph on
can
\emph default
 exist even when the denominator of a quotient approaches zero.
 This is not the same as division by zero, because we are considering 
\begin_inset Formula $\ec x\rightarrow a$
\end_inset

 and 
\emph on
not
\emph default
 
\begin_inset Formula $\ec x=a$
\end_inset

.
\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
You saw an example of this in a previous module, when we showed that
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\[
\ec\lim_{x\rightarrow0}\frac{x^{2}-x}{x}=-1
\]

\end_inset

 
\emph on
even though
\emph default
 
\begin_inset Formula $\ec\lim_{x\rightarrow0}x=0$
\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
Algebraically, we can show this as follows:
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\begin{align*}
\ec\lim_{x\rightarrow0}\frac{x^{2}-x}{x} & \ec=\lim_{x\rightarrow0}\frac{\cancel{x}(x-1)}{\cancel{x}}
\end{align*}

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Factored common 
\begin_inset Formula $x$
\end_inset

 in numerator.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Algebraically, we can show this as follows:
\begin_inset VSpace -1ex
\end_inset


\begin_inset Formula 
\begin{align*}
\dims\lim_{x\rightarrow0}\frac{x^{2}-x}{x} & \dims=\lim_{x\rightarrow0}\frac{\cancel{x}(x-1)}{\cancel{x}}\\
 & \ec=\lim_{x\rightarrow0}(x-1)\\
 & \ec=\lim_{x\rightarrow0}x-\lim_{x\rightarrow0}(-1)=-1.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Notice that we used the properties of
\begin_inset Newline newline
\end_inset

 (1) the sum of a limit,
\begin_inset Newline newline
\end_inset

 (2) the limit of a constant, and
\begin_inset Newline newline
\end_inset

 (3) the limit of 
\begin_inset Formula $x$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\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

<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
On the other hand, limits do not 
\emph on
have
\emph default
 to exist, either.
 If 
\begin_inset Formula $\ec f(x)=\frac{|x|}{x}$
\end_inset

, then
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0^{-}}f(x)=\lim_{x\rightarrow0^{-}}\frac{-x}{x}=-1}\ldots
\]

\end_inset

 
\end_layout

\begin_layout Standard
\align center
(If 
\begin_inset Formula $\ec x\rightarrow0^{-}$
\end_inset

, the value of 
\begin_inset Formula $\ec x$
\end_inset

 is negative, so 
\begin_inset Formula $\ec|x|=-x$
\end_inset

.
\begin_inset Newline newline
\end_inset

 This puzzles people who read it the first time,
\begin_inset Newline newline
\end_inset

 so think about it carefully---it is 
\emph on
not
\emph default
 a typo!) 
\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
On the other hand, limits do not 
\emph on
have
\emph default
 to exist, either.
 If 
\begin_inset Formula $\ec f(x)=\frac{|x|}{x}$
\end_inset

, then
\begin_inset Formula 
\[
{\dims\lim_{x\rightarrow0^{-}}f(x)=\lim_{x\rightarrow0^{-}}\frac{-x}{x}=-1}
\]

\end_inset

but
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0^{+}}f(x)=\lim_{x\rightarrow0^{+}}\frac{x}{x}=1}.
\]

\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
On the other hand, limits do not 
\emph on
have
\emph default
 to exist, either.
 If 
\begin_inset Formula $\ec f(x)=\frac{|x|}{x}$
\end_inset

, then
\begin_inset Formula 
\[
{\dims\lim_{x\rightarrow0^{-}}f(x)=\lim_{x\rightarrow0^{-}}\frac{-x}{x}=-1}
\]

\end_inset

but
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0^{+}}f(x)=\lim_{x\rightarrow0^{+}}\frac{x}{x}=1}.
\]

\end_inset

 So
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0^{-}}f(x)\neq\lim_{x\rightarrow0^{+}}f(x)}
\]

\end_inset

and there is no two-sided limit.
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace -2ex
\end_inset

When division by zero appears in a limit, 
\emph on
further investigation is needed
\emph default
.
\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\noindent
\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
We identified (and used!) a number of properties of the limit.
 These properties will be useful in future modules.
 
\end_layout

\begin_layout Itemize
We proved some properties using the precise definition of the limit (
\begin_inset Formula $\epsilon$
\end_inset

-
\begin_inset Formula $\delta$
\end_inset

).
 The others can also be proved, of course.
 
\end_layout

\begin_layout Itemize
If we are using the quotient property of limits and encounter division by
 zero, then further investigation is needed.
 A useful tool for such investigation is factoring.
 Another useful tool is analyzing the one-sided limits.
 
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