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\begin{document}
%MODULE 1: Epsilon-Delta limits


%USER DEFINED COMMANDS
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%INPUT THE INSTRUCTIONS PAGE 


%BUILD  TITLE PAGE 



\title{Limits at $\infty$}


\subtitle{We look at the long-term behavior of functions.}

\makebeamertitle

\lyxframeend{}\lyxframe{{Introduction}}

Up to now we have only considered limits as they approach \emph{a
fixed number} $\ec a$:
\[
\ec\lim_{x\rightarrow a}f(x).
\]
 Often we are interested as well in the \emph{long-term} behavior
of a function: \begin{question} 
\begin{itemize}
\item What happens to the $y$ values as $\ec x$ increases without bound?
That is, what is
\[
\ec\lim_{x\rightarrow\infty}f(x)?
\]
 
\item What happens to the $y$ values as $\ec x$ decreases without bound?
That is, what is
\[
\ec\lim_{x\rightarrow-\infty}f(x)?
\]
 
\end{itemize}
\noindent \end{question}


\lyxframeend{}\lyxframe{{Three sample problems}}

Let 
\begin{itemize}
\item ${\displaystyle {\ec f\left(x\right)=\frac{3x+1}{x}}}$, 
\item $\ec s(x)=\sin(x)$, and 
\item ${\displaystyle {\ec d\left(x\right)=\frac{1}{x}}}$. 
\end{itemize}
\noindent We want to estimate, or even evaluate,
\[
{\ec\lim_{x\rightarrow\pm\infty}f(x)}\quad\mathrm{and}\quad{\ec\lim_{x\rightarrow\pm\infty}s(x)}\quad\mathrm{and}\quad{\ec\lim_{x\rightarrow\pm\infty}d(x)},
\]
\emph{assuming they exist!}.


\lyxframeend{}\lyxframe{{The limit at infinity}}

Recall from previous lessons that: 
\begin{itemize}
\item the mathematical term \emph{limit} indicates the value a variable
or a function approaches; 
\item we can use numerical or graphical methods to estimate the value of
a limit, if it exists; and 
\item we can use the precise definition of a limit ($\varepsilon$-$\delta$)
to show rigorously that our estimate is correct, or that none exists. 
\end{itemize}

\lyxframeend{}\lyxframe{\noindent {Infinity is not a number!}}

\noindent It is important to keep in mind that 

\noindent \begin{center}
\emph{infinity is not a number!}
\par\end{center}

We cannot {}``substitute'' it in place of $\ec x$, add to it, subtract
from it, and so forth.

Infinity is a \emph{word} that we use as a shorthand for an \emph{idea}:
namely, increasing or decreasing without bound.


\lyxframeend{}\lyxframe{{SAGE worksheets}}

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


\lyxframeend{}\lyxframe{\noindent {First example: numerical}}
\begin{overlayarea}
{\textwidth}{6in}

\noindent Let's use all three methods on the first example,
\[
{\ec\lim_{x\rightarrow\pm\infty}f(x)}\quad\mathrm{where}\quad{\ec f(x)=\frac{3x+1}{x}}.
\]
 We will start with a numerical estimate: what happens to the values
of $\ec f$ as $\ec x$ increases without bound?


\pause{}


Since $\ec x$ increases without bound, we'll build a table of $x$-$y$
values where $\ec x$ increases. Feel free to pick grotesquely big
numbers!


\pause{}
\begin{onlyenv}%{}
<3-4>

\begin{center}
{\ec%
\begin{tabular}{c|c|c|c|c|c|c}
$x$  & 1  & 10  & 100  & 1000  & 10000  & 1,000,000,000 \tabularnewline
\hline 
$y$  & ?  & ?  & ?  & ?  & ?  & ? \tabularnewline
\end{tabular}}
\par\end{center}


\pause{}
\begin{onlyenv}%{}
<4>

\noindent Remember that you can use SAGE to pass quickly through the
computations: define a function $\ec f$, then evaluate $\ec f$ at
each point. You can also use a loop: {\ttcolor {[}f(a) for a in
{[}1,10,\ldots{}{]}{]}} as described in the module on intuitive
limits.

\end{onlyenv}%{}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<5>

\begin{center}
{\ec%
\begin{tabular}{c|c|c|c|c|c|c|c}
$x$  & 1.0  & 10.0  & 100.0  & 1000.0  & 10000.0  & 1,000,000,000.0  & {\hilite$\rightarrow\infty$}\tabularnewline
\hline 
$y$  & 4.0  & 3.1  & 3.01  & 3.001  & 3.0001  & 3.000000001  & {\hilite$\rightarrow3$}\tabularnewline
\end{tabular}}
\par\end{center}


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


\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{First example: graphical}}
\begin{overlayarea}
{\textwidth}{6in}

Let's use all three methods on the first example,
\[
\ec\lim_{x\rightarrow\pm\infty}f(x)\quad\mathrm{where}\quad f(x)=\frac{3x+1}{x}.
\]
Now let's try a graphical approach. Plot the graph of $\ec f(x)$
several times, with different maximum values of $\ec x$.


\pause{}
\begin{onlyenv}%{}
<2>
\end{onlyenv}%{}

If we try \texttt{\ttcolor plot(f,-10,10)} we see the following:
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_f_too_large} 
\end{figure}



\noindent From the graph we might guess that
\[
\ec\lim_{x\rightarrow\infty}f(x)=0.
\]
\emph{But this disagrees with our numerical estimate!}

\end{overlayarea}

\lyxframeend{}\lyxframe{{What happened?}}

Take a closer look at the graph: 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.3]{figures/plot_f_too_large} 
\end{figure}


\noindent Do you see how the $y$ values have such large magnitude?
They go up to 100-ish and down to -1900-ish.

To get a more accurate picture, we can use the \texttt{\ttcolor show()}
command with the \texttt{\ttcolor ymin} and \texttt{\ttcolor ymax}
attributes. Try typing this:

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

fplot.show(ymin=-10,ymax=10)


\lyxframeend{}\lyxframe{{First example: graphical}}

This new approach gives us a much better picture: 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_f_near_origin} 
\end{figure}


\noindent This new plot \emph{suggests} that the $y$ values approach
3.


\lyxframeend{}\lyxframe{{First example: graphical}}
\begin{overlayarea}
{\textwidth}{6in}

However, our graph isn't that convincing. It \emph{suggests} that
the $y$ values approach 3. But we're only going out to $\ec x=10$.
Lots of funny things can happen beyond $\ec x=10$. To get around
this, experiment with the $x$-interval in the plot.


\pause{}
\begin{onlyenv}%{}
<2>

We might, for example, try the interval $\ec[1,100]$: 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_f_0_100} 
\end{figure}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>

Or the interval $\ec[1,10000]$: 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_f_0_10000} 
\end{figure}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<4>

You can also shrink the $y$-interval in the \texttt{\ttcolor show()}
command to $\ec[2.9,3.1]$: 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_f_small_y_interval} 
\end{figure}



\noindent (Use {\ttcolor ymin=2.9,ymax=3.1} as options to {\ttcolor
show()}.)

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<5>

No matter how we change the graph, it \emph{appears} that the $y$
values are approaching $\ec y=3$ as $\ec x$ increases without bound.
Intuitively, we believe that
\[
\ec\lim_{x\rightarrow\infty}f(x)=3.
\]
(The same also appears to be true for $\ec-\infty$.)

\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{A precise approach!}}
\begin{overlayarea}
{\textwidth}{6in}

How can we \emph{convince} ourselves that a limit at infinity is $\ec L$?
We will need a precise definition for the limit at infinity.


\pause{}
\begin{onlyenv}%{}
<2-3>

For any measurable distance $\ec\varepsilon$, we want to know that
eventually the $y$ value of $\ec f(x)$ is indistinguishable from
the limit $\ec3$.


\pause{}
\begin{onlyenv}%{}
<3>

In other words, we want 
\begin{itemize}
\item for all $\ec\varepsilon>0$, 
\item sooner or later, \emph{every} $x$ value 
\item produces a $y$ value that is closer to $\ec L$ than $\ec\varepsilon$. 
\end{itemize}
\end{onlyenv}%{}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<4->

Rethinking our wording again,
\begin{definition}%{}
[(Limit at $\infty$)] $\ec\lim_{x\rightarrow\infty}f(x)=L$ if 
\begin{itemize}
\item for all $\ec\varepsilon>0$, 
\item there exists $\ec N$ such that for \emph{every} $\ec x>N$ 
\item $\ec|f(x)-L|<\varepsilon$. 
\end{itemize}
\end{definition}%{}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<5>

Let's try this with the example, using $\ec\varepsilon=0.01$. We
want
\begin{align*}
{\ec\left|f(x)-L\right|} & {\ec<0.01}.
\end{align*}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<6>

Let's try this with the example, using $\ec\varepsilon=0.01$. We
want
\begin{align*}
{\dims\left|f(x)-L\right|} & {\dims<0.01}\\
{\ec\left|\frac{3x+1}{x}-3\right|} & {\ec<0.01}.
\end{align*}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<7>

Let's try this with the example, using $\ec\varepsilon=0.01$. We
want
\begin{align*}
{\dims\left|\frac{3x+1}{x}-3\right|} & {\dims<0.01}\\
{\ec-0.01<\frac{3x+1}{x}-3} & {\ec<0.01}.
\end{align*}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<8>

Let's try this with the example, using $\ec\varepsilon=0.01$. We
want
\begin{align*}
{\dims-0.01<\frac{3x+1}{x}-3} & {\dims<0.01}\\
{\ec3-0.01<\frac{3x+1}{x}} & {\ec<3+0.01}.
\end{align*}
 \begin{explanation}Added $3$ to both sides.\end{explanation}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<9>

Let's try this with the example, using $\ec\varepsilon=0.01$. We
want
\begin{align*}
{\dims3-0.01<\frac{3x+1}{x}} & {\dims<3+0.01}\\
{\ec x(3-0.01)<3x+1} & {\ec<x(3+0.01)}.
\end{align*}
 \begin{explanation}Multiplied $x$ to both sides.\\
 Since $x>0$ as $x\rightarrow\infty$,\\
 the direction of the inequalities does not change.\end{explanation}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<10>

Let's try this with the example, using $\ec\varepsilon=0.01$. We
want
\begin{align*}
{\dims x(3-0.01)<3x+1} & {\dims<x(3+0.01)}\\
{\ec-0.01x<1} & {\ec<0.01x}.
\end{align*}
 \begin{explanation}Subtracted $3x$ from both sides.\end{explanation}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<11>

Let's try this with the example, using $\ec\varepsilon=0.01$. We
want
\begin{align*}
{\dims-0.01x<1} & {\dims<0.01x}\\
{\ec-0.01x<1} & \mathrm{\ and\ }{\ec1<0.01x}.
\end{align*}
 \begin{explanation}Meaning of the inequality.\end{explanation}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<12->

Let's try this with the example, using $\ec\varepsilon=0.01$. We
want
\begin{align*}
{\dims-0.01x<1} & \mathrm{\ and\ }{\dims1<0.01x}\\
\mathrm{so\ we\ want\ }\quad{\ec x>-\frac{1}{0.01}} & \mathrm{\ and\ }{\ec x>\frac{1}{0.01}}.
\end{align*}
 The larger of these is $\ec1/0.01=100$, so we want $\ec x>100$.

\end{onlyenv}%{}

\pause{}


\noindent In fact, $\ec f(101)\approx3.0099$ and $\ec\left|3.0099-3\right|=.0099<.01$.

\end{overlayarea}

\lyxframeend{}\lyxframe{{Second example}}
\begin{overlayarea}
{\textwidth}{6in}

Let's look at $\ec s(x)=\sin(x)$.


\pause{}
\begin{onlyenv}%{}
<2>

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


\begin{center}
{\ec%
\begin{tabular}{c|c|c|c|c|c|c}
$x$  & 1.0  & 10.0  & 100.0  & 1000.0  & 10000.0  & 1,000,000,000.0 \tabularnewline
\hline 
$y$  & 0.8415  & -0.5540  & -0.5063  & 0.8269  & -0.3056  & 0.5458 \tabularnewline
\end{tabular}}
\par\end{center}


\noindent \ldots{}we don't see any trend to \emph{one} special number.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>

If we plot the function for intervals with larger and larger $x$
values\ldots{} 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_s_near_origin} 
\end{figure}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<4>

If we plot the function for intervals with larger and larger $x$
values\ldots{} 
\begin{figure}[h!]
 

\centering{}\includegraphics[scale=0.2]{figures/plot_s_0_100} 
\end{figure}



\noindent \ldots{}we see that $\ec s$ oscillates between -1 and
1. Still there is no trend to \emph{one} special value.


It looks as if there is no limit.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<5>

Recall the precise definition.
\begin{definition}%{}
[(Limit at $\infty$)] $\ec\lim_{x\rightarrow\infty}f(x)=L$ if 
\begin{itemize}
\item for all $\ec\varepsilon>0$, 
\item there exists $\ec N$ such that for \emph{every} $\ec x>N$ 
\item $\ec|f(x)-L|<\varepsilon$. 
\end{itemize}
\end{definition}%{}

In our case, 
\begin{itemize}
\item For any hypothetical limit $\ec L$, 
\item if $\ec\varepsilon=0.9$, 
\item then regardless of the value of $\ec N$, 
\item we can find infinitely many $x$ values 
\item such that $\ec|\sin(x)-L|>\varepsilon$. 
\end{itemize}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<6>

For any hypothetical limit $L$, if $\varepsilon=0.9$, then regardless
of the value of $N$, we can find infinitely many $x$ values such
that $|\sin(x)-L|>\varepsilon$.


For example, if $\ec L=0$, then $\ec s\left(\frac{\pi}{2}\right)=1$,
$\ec s\left(\frac{5\pi}{2}\right)=1$, $\ec s\left(\frac{9\pi}{2}\right)=1$,
\ldots{}. This pattern continues for ever, so the limit of the $y$
values does not approach 0 even though $\ec y=0$ infinitely many
times at $\ec x=0,\pi,2\pi,\ldots$.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<7>

For any hypothetical limit $L$, if $\varepsilon=0.9$, then regardless
of the value of $N$, we can find infinitely many $x$ values such
that $|\sin(x)-L|>\varepsilon$.


We could play this same game with \emph{any} value of $\ec L$. (Try
it!) So,
\[
{\ec\lim_{x\rightarrow\infty}s(x)\mathrm{\ does\ not\ exist}}.
\]


\end{onlyenv}%{}
\end{overlayarea}

\lyxframeend{}\lyxframe{{Third example}}
\begin{overlayarea}
{\textwidth}{6in}

Finally, we'll look at $\ec d(x)=1/x$.


\pause{}
\begin{onlyenv}%{}
<2>

This is an important and useful function. Along with its limits, $\ec d(x)$
recurs quite a bit in Calculus.

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3->

Use the numerical approach to fill in the question marks, and obtain
an estimate for $\ec\lim_{x\rightarrow\infty}1/x$.
\begin{onlyenv}%{}
<3>
\end{onlyenv}%{}

\begin{center}
{\ec%
\begin{tabular}{c|c|c|c|c|c|c}
$x$  & 1.0  & 10.0  & 100.0  & 1000.0  & 10000.0  & 1,000,000,000.0 \tabularnewline
\hline 
$y$  & ?  & ?  & ?  & ?  & ?  & ? \tabularnewline
\end{tabular}}
\par\end{center}


The values \emph{should} seem to approach a number.


\pause{}
\begin{onlyenv}%{}
<4>

\begin{center}
{\ec%
\begin{tabular}{c|c|c|c|c|c|c}
$x$  & 1.0  & 10.0  & 100.0  & 1000.0  & 10000.0  & 1,000,000,000.0 \tabularnewline
\hline 
$y$  & 1.0  & 0.1  & 0.01  & 0.001  & 0.0001  & 0.000000001 \tabularnewline
\end{tabular}}
\par\end{center}


The values \emph{should} seem to approach a number.

\end{onlyenv}%{}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<6->
\end{onlyenv}%{}

\begin{center}
{\ec%
\begin{tabular}{c|c|c|c|c|c|c}
$x$  & 1.0  & 10.0  & 100.0  & 1000.0  & 10000.0  & 1,000,000,000.0 \tabularnewline
\hline 
$y$  & 1.0  & 0.1  & 0.01  & 0.001  & 0.0001  & 0.000000001 \tabularnewline
\end{tabular}}
\par\end{center}


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


\pause{}


You should have obtained the estimate 
\[
{\ec\lim_{x\rightarrow\infty}\frac{1}{x}=0}.
\]
Let's check this using the precise definition!

\end{overlayarea}

\lyxframeend{}\lyxframe{{Third example: precise}}
\begin{overlayarea}
{\textwidth}{6in}

For any $\ec\varepsilon$, we want to find a value of $\ec N$ such
that if $\ec x>N$ then $\ec|1/x-0|<\varepsilon$. That is,
\begin{onlyenv}%{}
<1>
\begin{align*}
{\ec\left|\frac{1}{x}\right|} & {\ec<\varepsilon}.
\end{align*}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<2>
\begin{align*}
\ec{\dims\left|\frac{1}{x}\right|} & {\dims<\varepsilon}\\
{\ec-\varepsilon<\frac{1}{x}} & {\ec<\varepsilon}.
\end{align*}
 \begin{explanation}Property of this absolute value inequality.\end{explanation}
\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<3>

\noindent 
\begin{align*}
\ec{\dims-\varepsilon<\frac{1}{x}} & {\dims<\varepsilon}\\
{\ec-\varepsilon x<1} & {\ec<\varepsilon x}.
\end{align*}
 \begin{explanation}Multiplied both sides by $x$.\\
 Since $x>0$ as $x\rightarrow\infty$,\\
 we don't have to change the direction of the inequality.\end{explanation}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<4>

\noindent 
\begin{align*}
{\dims-\varepsilon x<1} & {\dims<\varepsilon x}\\
{\ec-\varepsilon x<1} & \mathrm{\ and\ }{\ec1<\varepsilon x}.
\end{align*}
 \begin{explanation}Meaning of this inequality.\end{explanation}

\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<5>

\begin{align*}
\ec{\dims-\varepsilon x<1} & \mathrm{\dims\ and\ }{\dims1<\varepsilon x}\\
\mathrm{so\ }{\ec x>-\frac{1}{\varepsilon}} & \mathrm{\ and\ }{\ec x>\frac{1}{\varepsilon}}.
\end{align*}


\end{onlyenv}%{}

\pause{}
\begin{onlyenv}%{}
<6>

\noindent 
\begin{align*}
\ec{\dims-\varepsilon x<1} & {\dims<\varepsilon x}\\
\mathrm{so\ }{\ec x>-\frac{1}{\varepsilon}} & \mathrm{\ and\ }{\ec x>\frac{1}{\varepsilon}}.
\end{align*}
 The larger of these two values is $\ec1/\varepsilon$.

\end{onlyenv}%{}

\pause{}


\noindent If, for example, $\ec\varepsilon=0.1$, then every value
of $\ec x$ that is \emph{greater} than $\ec1/\varepsilon=10$ would
satisfy $\ec|d(x)|<\varepsilon$. (\emph{Check} it!) This proves that
$\ec\lim_{x\rightarrow\infty}d(x)=0$.

\end{overlayarea}

\lyxframeend{}\lyxframe{{BIG-TIME FACT}}

As mentioned, the fact we just proved is something you should remember
for future reference:
\begin{theorem}%{}
\[
{\ec\lim_{x\rightarrow\infty}\frac{1}{x}=0}.
\]

\end{theorem}%{}

\lyxframeend{}\lyxframe{\noindent {The other direction}}

\noindent In all the examples we looked at in this module, we focused
on \emph{positive} infinity. Similar arguments can be made for limits
at \emph{negative} infinity. The definition that you would use is
\begin{definition}%{}
[(Limit at $-\infty$)] $\ec\lim_{x\rightarrow-\infty}f(x)=L$ if 
\begin{itemize}
\item for all $\ec\varepsilon>0$, 
\item there exists $\ec N$ such that for \emph{every} $\ec x<N$ 
\item $\ec|f(x)-L|<\varepsilon$. 
\end{itemize}
\end{definition}%{}
\noindent In this case, multiplying or dividing the inequalities by
$\ec x$ would require you to change the direction of the inequality.


\lyxframeend{}\lyxframe{Conclusion}
\begin{itemize}
\item When we want to analyze how the $y$ values of a function $\ec f$
behave when $\ec x$ increases (or decreases) without bound, we talk
about this as \emph{the limit of $\ec f$ as $\ec x$ approaches infinity
(or negative infinity)}. 
\item This limit is defined as\setlength{\shadeboxwidth}{4in}\vspace{-2ex}


\begin{definition}%{}
[(Limit at $\infty$)] $\ec\lim_{x\rightarrow\infty}f(x)=L$ if 
\begin{itemize}
\item for all $\ec\varepsilon>0$, 
\item there exists $\ec N$ such that for \emph{every} $\ec x>N$ 
\item $\ec|f(x)-L|<\varepsilon$. 
\end{itemize}
\end{definition}%{}
\item As with limits at finite values, there can only be one value for such
a limit, and they do not always exist. 
\item All three methods of estimating or evaluating a limit --- numerical,
graphical, precise --- can be used at infinity. 
\item An important limit at infinity to remember is $\ec\lim_{x\rightarrow\infty}1/x=0$. 
\end{itemize}

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