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%MODULE 1: Squeeze Theorem
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\begin_layout Title
The Squeeze Theorem
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\begin_layout Subtitle
Sometimes, the limit of a function 
\begin_inset Formula $\ec f$
\end_inset

 is difficult to evaluate using direct methods, but we can compare its values
 to those of other functions.
 Using the other functions as a guide, we can sometimes determine the limit
 of 
\begin_inset Formula $f$
\end_inset

.
 
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Sample problem
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 The problem:
\series default
 Evaluate 
\begin_inset Formula 
\[
\ec\lim_{x\rightarrow0}x\,\sin\left(\frac{1}{x}\right).
\]

\end_inset


\end_layout

\begin_layout Pause

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\begin_layout Standard
\noindent
How would you start the problem?
\end_layout

\begin_layout Standard
Take a look at the graph of 
\begin_inset Formula $\ec{\displaystyle x\sin\left(\frac{1}{x}\right)}$
\end_inset

.
 
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\begin_inset Graphics
	filename figures/basic_function.png
	lyxscale 20
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\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
 We see that 
\begin_inset Formula $\ec f$
\end_inset

 
\emph on
appears
\emph default
 to approach zero as 
\begin_inset Formula $\ec x$
\end_inset

 approaches zero.
 
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{
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The problem with geometry
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\begin_layout Standard
On the other hand, we can't always rely on geometric intuition.
 For instance, 
\begin_inset Formula $\ec{\displaystyle (x+0.01)\sin\left(\frac{1}{x}\right)}$
\end_inset

 also appears to approach zero as 
\begin_inset Formula $\ec x$
\end_inset

 approaches zero: 
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\begin_layout Plain Layout
\begin_inset Caption

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 We see that 
\begin_inset Formula $\ec{\displaystyle (x+0.01)\sin\left(\frac{1}{x}\right)}$
\end_inset

 
\emph on
appears
\emph default
 to approach zero as 
\begin_inset Formula $\ec x$
\end_inset

 approaches zero.
 
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\noindent
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{
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More of a problem with geometry
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\begin_layout Standard
\noindent
If we zoom in on the 
\begin_inset Formula $\ec x$
\end_inset

-axis, we see that there might 
\emph on
not
\emph default
 be a limit at 
\begin_inset Formula $\ec x=0$
\end_inset

.
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\noindent
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\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
 Here 
\begin_inset Formula $\ec{\displaystyle (x+0.01)\sin\left(\frac{1}{x}\right)}$
\end_inset

 
\emph on
appears
\emph default
 to wiggle wildly as 
\begin_inset Formula $\ec x$
\end_inset

 approaches zero.
 
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\end_inset


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In this case there is no limit at 
\begin_inset Formula $\ec x=0$
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.
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Squeezing a function
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name "pg:squeezing"

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 In many cases, a function is 
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{
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 trapped
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}
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 between two other functions.
 Another way of saying this is that it is 
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{
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 squeezed
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}
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 between two functions.
 For example, 
\begin_inset Formula $\ec{\displaystyle f(x)=x\sin\left(\frac{1}{x}\right)}$
\end_inset

 is trapped between the two functions 
\begin_inset Formula $\ec g(x)=x$
\end_inset

 and 
\begin_inset Formula $\ec h(x)=-x$
\end_inset

.
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	filename figures/basic_function_squeezed.png
	lyxscale 20
	scale 30

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\begin_inset Caption

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 The dashed red lines are 
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.
 
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In this case, we have 
\begin_inset Formula $\ec h(x)\le f(x)\le g(x)$
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 for all 
\begin_inset Formula $\ec x$
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.
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{
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How do we know it is squeezed?
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}
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\emph on
How do we know that the function is squeezed?
\emph default
 All that we saw there was a graph; zooming in sufficiently might show that
 something strange is going on.
\end_layout

\begin_layout Pause

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\begin_layout Standard
We can use simple facts from trigonometry to analyze the function.
 Start with 
\begin_inset Formula 
\[
{\ec-1\leq\sin\left(\frac{1}{x}\right)\leq1\quad\forall x\in\mathbb{R}}.
\]

\end_inset


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\begin_layout Pause

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We want to look at 
\begin_inset Formula $\ec x\sin(\frac{1}{x})$
\end_inset

, not 
\begin_inset Formula $\sin(\frac{1}{x})$
\end_inset

, so multiply all sides by 
\begin_inset Formula $\ec x$
\end_inset

.
 If 
\begin_inset Formula $\ec x{\ecgr\geq}0$
\end_inset

, we have 
\begin_inset Formula 
\[
{\ec-x{\ecgr\leq}x\sin\left(\frac{1}{x}\right){\ecgr\leq}1\quad\forall x\geq0}.
\]

\end_inset


\end_layout

\begin_layout Pause

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\begin_layout Standard
We want to look at 
\begin_inset Formula $\ec x\sin(\frac{1}{x})$
\end_inset

, not 
\begin_inset Formula $\sin(\frac{1}{x})$
\end_inset

, so multiply all sides by 
\begin_inset Formula $\ec x$
\end_inset

.
 If 
\begin_inset Formula $\ec x<0$
\end_inset

, we change the direction of the inequalities, and 
\begin_inset Formula 
\[
{\ec-x{\ecgr\geq}x\sin\left(\frac{1}{x}\right){\ecgr\geq1}\quad\forall x\geq0}.
\]

\end_inset


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{
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How does this help with the limits?
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\begin_layout Standard
\noindent
At this point we describe a well-known result in mathematics.
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\begin_layout Theorem
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[
\end_layout

\end_inset

(The Squeeze Theorem)
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]
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 Let 
\begin_inset Formula $\ec f$
\end_inset

, 
\begin_inset Formula $\ec g$
\end_inset

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

 be functions on 
\begin_inset Formula $\ec I=(a,b)$
\end_inset

 and 
\begin_inset Formula $\ec c\in I$
\end_inset

.
 If 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec f(x)$
\end_inset

 lies between 
\begin_inset Formula $\ec g(x)$
\end_inset

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

 for all 
\begin_inset Formula $\ec x\in I$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec{\displaystyle L=\lim_{x\rightarrow c}g(x)=\lim_{x\rightarrow c}h(x)}$
\end_inset

, 
\end_layout

\begin_layout Standard
then 
\begin_inset Formula $\ec{\displaystyle \lim_{x\rightarrow c}f(x)=L}$
\end_inset

.
 
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\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
Read this carefully.
 Try to identify which functions from the previous slide correspond to 
\begin_inset Formula $\ec f$
\end_inset

, 
\begin_inset Formula $\ec g$
\end_inset

, 
\begin_inset Formula $\ec h$
\end_inset

, 
\begin_inset Formula $\ec c$
\end_inset

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

.
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{
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Applying the theorem to our example
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\begin_deeper
\begin_layout Theorem
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[
\end_layout

\end_inset

(The Squeeze Theorem)
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]
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\end_inset

 Let 
\begin_inset Formula $\ec f$
\end_inset

, 
\begin_inset Formula $\ec g$
\end_inset

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

 be functions on 
\begin_inset Formula $\ec I=(a,b)$
\end_inset

 and 
\begin_inset Formula $\ec c\in I$
\end_inset

.
 If 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec f(x)$
\end_inset

 lies between 
\begin_inset Formula $\ec g(x)$
\end_inset

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

 for all 
\begin_inset Formula $\ec x\in I$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec{\displaystyle L=\lim_{x\rightarrow c}g(x)=\lim_{x\rightarrow c}h(x)}$
\end_inset

, 
\end_layout

\begin_layout Standard
then 
\begin_inset Formula $\ec{\displaystyle \lim_{x\rightarrow c}f(x)=L}$
\end_inset

.
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<1>
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\begin_deeper
\begin_layout Standard
In our example, we want to show that 
\begin_inset Formula $\ec\lim_{x\rightarrow0}x\sin(1/x)=0$
\end_inset

.
 We want to use the fact that 
\begin_inset Formula $\ec x\sin(1/x)$
\end_inset

 always lies between 
\begin_inset Formula $\ec x$
\end_inset

 and 
\begin_inset Formula $\ec-x$
\end_inset

.
\end_layout

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<2>
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These values correspond (
\begin_inset Formula ${\ec\mapsto}$
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) in the following way to the terms of the 
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 Squeeze Theorem
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}
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\end_inset

: 
\begin_inset Formula 
\begin{eqnarray*}
\ec f(x) & {\ecgr\mapsto} & {\ec x\sin\left(\frac{1}{x}\right)},\\
\ec g(x) & {\ecgr\mapsto} & {\ec x},\\
\ec h(x) & {\ecgr\mapsto} & {\ec-x},\\
\ec c & {\ecgr\mapsto} & {\ec0},\\
\ec L & {\ecgr\mapsto} & {\ec0}.
\end{eqnarray*}

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<3->
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\begin_deeper
\begin_layout Standard
Since 
\begin_inset Formula $\ec\lim_{x\rightarrow0}x=0$
\end_inset

 and 
\begin_inset Formula $\ec\lim_{x\rightarrow0}(-x)=0$
\end_inset

, what can we conclude about 
\begin_inset Formula $\ec\lim_{x\rightarrow0}x\sin(1/x)$
\end_inset

?
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<4>
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\begin_deeper
\begin_layout Standard
By the Squeeze Theorem, 
\begin_inset Formula $\ec\lim_{x\rightarrow0}x\,\sin(1/x)=0$
\end_inset

.
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\end_deeper
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{
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The bad example
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\begin_deeper
\begin_layout Standard
\noindent
What about that function with an infinite wiggle? In this case, 
\begin_inset Formula 
\[
\ec x+0.01\geq\left(x+0.01\right)\sin\left(\frac{1}{x}\right)\geq-(x+0.01).
\]

\end_inset


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<1>
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\begin_deeper
\begin_layout Standard
\noindent
\begin_inset Float figure
placement h!
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\align center
\begin_inset Graphics
	filename figures/deceptive_function_not_squeezed.png
	lyxscale 20
	scale 30

\end_inset

 
\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
 The dashed red lines correspond to 
\begin_inset Formula $\ec y=\pm(x+0.01)$
\end_inset

.
 It may look as if the function crosses those lines, but it does not.
 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
Doesn't the Squeeze Theorem apply here, too? Not at 
\begin_inset Formula $\ec x=0$
\end_inset

! Why not?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Here, as 
\begin_inset Formula $\ec x\rightarrow0$
\end_inset

, 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0}(x+0.01)=0.01},
\]

\end_inset

whereas 
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0}(x-0.01)=-0.01}.
\]

\end_inset

Since the limits are unequal, 
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 the hypothesis of the Squeeze Theorem are unsatisfied
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\end_inset

.
 We cannot conclude anything about 
\begin_inset Formula $\ec{\displaystyle \lim_{x\rightarrow0}(x+0.01)\sin\left(\frac{1}{x}\right)}$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
The bad example: putting it together
\end_layout

\begin_layout Standard
\noindent
Geometrically, we see this in graph of the function 
\begin_inset Formula $\ec y=(x+0.01)\sin(1/x)$
\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/deceptive_function_not_squeezed.pdf
	lyxscale 50
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The limits 
\begin_inset Quotes eld
\end_inset

look
\begin_inset Quotes erd
\end_inset

 as if they should be 
\begin_inset Formula $\ec{\displaystyle \lim_{x\rightarrow0}(x+0.01)=0.01}$
\end_inset

, and 
\begin_inset Formula $\ec{\displaystyle \lim_{x\rightarrow0}(x-0.01)=-0.01}$
\end_inset

, as explained on the previous slide.
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Finding functions to squeeze another
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The 
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 Squeeze Theorem
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\end_inset

 allows you to apply your understanding of a limit in many places.
 With the sine and cosine functions, you can usually Start with 
\begin_inset Formula 
\[
{\ec-1\leq\sin(\mathrm{\mc\/anything})\leq1},\hspace{1em}*{1em}\mbox{ and}\hspace{1em}*{1em}{\ec-1\leq\cos(\mathrm{\mc\/anything})\leq1}.
\]

\end_inset

 Then build this relationship up to the point that we are investigating,
 as we did on 
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pg:squeezing
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.
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\begin_layout BeginFrame
\noindent
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The process of estimation
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\begin_layout Standard
\noindent
In finding functions 
\begin_inset Formula $\ec g(x)$
\end_inset

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

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

 satisfies 
\begin_inset Formula 
\[
{\ec g(x)\leq f(x)\leq h(x),\qquad\forall x},
\]

\end_inset

 we have managed to 
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 bound 
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 the behaviour of 
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.
 In this case 
\begin_inset Formula $\ec g(x)$
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 is a 
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 lower bound
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 and 
\begin_inset Formula $\ec h(x)$
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 is an 
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 upper bound
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 for 
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.
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\begin_layout Standard
The process of replacing a known function with an estimate, usally a lower
 or upper bound, or both, is know as 
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 estimation
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.
 A large part of mathematical analysis consists of trying to find estimates
 that desribe the behaviour of very complicated functions.
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Conclusion
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\begin_layout Itemize
The 
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 Squeeze Theorem
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 allows us to evaluate easily the values of many limits that are otherwise
 difficult.
 
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\begin_layout Itemize
To use the Squeeze Theorem, you need to identify upper and lower bounds
 for a function.
 Those upper and lower bounds must have the same limit at the point in question.
 
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\end_body
\end_document
