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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Precise Definition of a Limit}

We now have to make precise, mathematical sense out of the statement \[
\lim _{x\rightarrow a}f(x)=L.\]
 

Recall the informal definition: 

\begin{defn}
We say \[
\lim _{x\rightarrow a}f(x)=L\]
 if we can make \( f(x) \) \textit{as close as we need} to \( L \), just by
taking \( x \) \textit{close enough} to \( a \). 
\end{defn}
In order to make that precise, we need first to figure out how to quantify how
close \( f(x) \) is to \( L \). The distance between them is \( |f(x)-L| \).
So, how close do we need it? Say within \( \epsilon  \), which is some positive
number. (\emph{All} calculus texts \emph{always} use the letter \( \epsilon  \)
for this.) So, we need to be able to arrange \( |f(x)-L|<\epsilon  \), just
by taking \( x \) \textit{close enough} to \( a \). The distance from \( x \)
to \( a \) is \( |x-a| \). \textit{Close enough} should be some (small) positive
distance \( \delta  \). So, the  definition above becomes: 

\begin{defn}
Let \( f(x) \) be defined in a neighborhood of \( a \), except perhaps at
\( a \) itself. That is, assume for some \( c<a \) and \( b>a \), \( f \)
is  defined on the intervals \( (c,a) \) and \( (a,b) \). Then, we say that
\[
\lim _{x\rightarrow a}f(x)=L\]
 if, for every \( \epsilon >0 \), there is a \( \delta >0 \) so that, if  \( 0<|x-a|<\delta  \),
then \( |f(x)-L|<\epsilon  \). 
\end{defn}
\vspace{0.3cm}
{\par\centering \includegraphics{epsilon-delta.ps} \par}
\vspace{0.3cm}

\begin{rem}
The \( \epsilon  \) is {}``as close as you need{}'', and the \( \delta  \)
is {}``close enough{}''. The way we usually phrase it is to say that \textit{someone
else}, your \textit{opponent} in the game, chooses \( \epsilon  \), and you
then have to choose a good enough \( \delta  \) to win.
\end{rem}
\begin{note}
Notice how I snuck that \( 0<|x-a| \) in there. That just means  that \( x\neq a \).
Remember, in limits we don't care how, or if, \( f(a) \) is  defined, we want
to know what \( f(x) \) is tending towards as \( x\rightarrow a \). We want
to know how \( f(a) \) \textit{should be} defined. 
\end{note}
One-sided limits are handled the same way, except that \( 0<|x-a|<\delta  \)
is replaced by \( a<x<a+\delta  \) for the right-hand limit \[
\lim _{x\rightarrow a+}f(x),\]
 or \( a-\delta <x<a \) for the left-hand limit \[
\lim _{x\rightarrow a+}f(x).\]



\subsubsection{Precise definition of limits at \protect\( \infty \protect \)}

To begin with, recall the informal definition of the situation  \[
\lim _{x\rightarrow \infty }f(x)=\infty .\]
 In that case, we said informally that ``as \( x \) gets larger and larger,
so does \( f(x) \)''. What does that mean? It means that, \textit{no matter
how large} you insist that \( f(x) \) be, I can arrange that \textit{just}
by taking \( x \) \textit{large enough}.  

Now. let's make it precise. How large do you want me to make \( f(x) \)? It
would be some number, \( E \). I have to make \( f(x) \) larger than that,
at least. But, I'm only allowed to do that by taking \( x \) large enough,
say \( x>D \). If the limit  \[
\lim _{x\rightarrow \infty }f(x)=\infty \]


is true, I have to be able to do this, no matter what. Here goes: 

\begin{defn}
Assume that, for some number \( c \), \( f(x) \) is defined  for all \( x>c \).
Then, we say that \[
\lim _{x\rightarrow \infty }f(x)=\infty \]
 if, for each number \( E \), there is a number \( D \) so that, if \( x>D \),
then \( f(x)>E \). 
\end{defn}

\subsubsection{Finite limits as \protect\( x\rightarrow \infty \protect \) }

A more subtle case is the informal statement of the condition that \[
\lim _{x\rightarrow \infty }f(x)=L.\]
  There, we say informally that ``in order to make \( f(x) \) as close as we
need to  \( L \), we just need to take \( x \) large enough''. 

Assume that, for some number \( c \), \( f(x) \) is defined  for all \( x>c \).
Then, we say that \[
\lim _{x\rightarrow \infty }f(x)=L\]
 if, for each number \( \epsilon >0 \), there is a number \( D \) so that,
if \( x>D \),  then \( |f(x)-L|<\epsilon  \). 


\subsubsection{Infinite limits as \protect\( x\rightarrow a\protect \)}

\begin{defn}
A function \( f(x) \) has an infinite limit as \( x \) approaches \( a \)
(or has a vertical asymptote at \( x=a \)), \[
\lim _{x\rightarrow a}f(x)=\infty ,\]
if, for each \( E>0 \), there is a \( \delta >0 \) so that whenever \( 0<|x-a|<\delta  \),
\( |f(x)|>E \). 
\end{defn}
You can figure out what the analogue would be for one-sided limits, or for limits
going specifically to \( +\infty  \), \( -\infty  \), or both ways, \( \pm \infty  \).
Of course, this last case is really the same as the definition we just gave.


\subsection*{Examples}

\begin{example}
Show that \[
\lim _{x\rightarrow 1}3x+5=8.\]
 
\end{example}

\subsubsection*{Solution}

\begin{example}
Show that \[
\lim _{x\rightarrow 2}x^{2}+3x=10.\]
 
\end{example}

\subsubsection*{Solution}

\begin{example}
Show that \[
\lim _{x\rightarrow 3}\frac{1}{x}=\frac{1}{3}.\]
 
\end{example}

\subsubsection*{Hints}

\begin{example}
Show that \[
\lim _{x\rightarrow \infty }\sqrt{x}=\infty .\]

\end{example}

\subsubsection*{Solution }

\begin{example}
Show that  \[
\lim _{x\rightarrow \infty }\frac{2x+3}{x-1}=2.\]

\end{example}

\subsubsection*{Hints}


\subsection*{Exercises}

\begin{xca}
Find a number \( \delta  \) sufficiently small so that the distance from \( f(x)=2x^{2}+3x-1 \)
to \( 4 \) is less than \( 1/100 \) if \( |x-1|<\delta  \). 

\[
\delta =\]

\end{xca}
%

\begin{xca}
Show that \[
\lim _{x\rightarrow \infty }(x/2+\sin x)=\infty .\]

\end{xca}

\subsubsection*{Answer}



\begin{xca}
Show that  \[
\lim _{x\rightarrow \infty }\frac{x^{2}+3x+2}{x^{2}-1}=2.\]
 
\end{xca}

\subsubsection*{Answer}
\end{document}
