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Recitation 12   \hfill \course\\
6 October 1998  \hfill \semester \\

\medskip

{\bf Reminders}: 
1. A notebook is on reserve in the math library for this course. This
notebook  contains tests from last year and  solutions to tests,
homework, quizzes and 
projects. The math library is located in the basement of the Patterson
Office  Tower. 

2. The next homework is due on Wednesday, 7 October. The assignment is
\S2.3 \#12, 20  \S2.4 \#6, 14.

3.  We are  one lecture  behind the schedule which was handed out for
section 2.  Thus, you will probably not get very far in this
worksheet on Tuesday. Section 2.6 is fairly short, thus I expect that we will
catch up by the end of this week. 




\medskip
 
Below is a selection of problems related to sections 2.5.
These problems will not be collected or graded. However, you should
understand how to work each of these problems.  You should begin
working on these problems in groups in recitation. You will probably
want to finish these problems outside of class. 
 If you have questions,
please ask your TA or instructor. If you find a problem difficult,
consider working similar problems from the text for additional practice.

\begin{enumerate}

\item (Algebra review) Obtain a common denominator and simplify the
numerator.
$$
\frac 1 {a+b} - \frac 1 a.
$$

\item (Algebra review) Obtain a common denominator and simplify the
numerator.
$$
\frac 1 {(a+b)^2} - \frac 1 {a^2}.
$$

\item Work the following exercises from Section 2.5.  \#3, 5, 7, 15,
17, 19, 21, 25, 29, 33, 35, 41.

\item What is $\infty - \infty$?    We have not defined
arithmetic with the symbol $\infty$. One way to carry out
arithmetic with infinity is to find a function $f(x)$ so that
$\lim_{x\rightarrow a} f(x) = \infty$. Then carry out
arithmetic with the  function $f(x)$  and take the limit.  Sometimes
this works well. For example, it can be seen that $\infty + \infty =
\infty$ or that for any real number $c$, $\infty +
c=\infty$. Sometimes, however, this does not work. The problem is that if
you make different choices for the function $f$, then you might get
different answers for $\infty-\infty$.  In this problem, you are asked
to produce examples 
to show that we cannot make a sensible definition of
$\infty-\infty$.  

This also explains why I insist that if a limit is $\infty$, then it
is undefined (but is $\infty$). This way, you will not be tempted to
apply the limit laws to limits which take the value $\infty$. 

For the problems below, let $f(x) = 1/x^2$.

\begin{enumerate}  

\item Consider $f(x)$, $g(x)=f(x)$ and $f(x)-g(x)$ and compute
$$
\lim _{x\rightarrow 0 } f(x), \qquad \lim _{x\rightarrow 0 } g(x),
\qquad
\lim _{x\rightarrow 0 } f(x)-g(x).$$

Based on your answers, what should $\infty - \infty$ be.


\item Consider $f(x)$, $g(x)= 73  +f(x)$ and $f(x)-g(x)$ and compute
$$
\lim _{x\rightarrow 0 } f(x), \qquad \lim _{x\rightarrow 0 } g(x),
\qquad
\lim _{x\rightarrow 0 } f(x)-g(x).$$

Based on your answers, what should $\infty - \infty$ be.

\item Can you give an example where it appears that $\infty
-\infty$ should equal 24?

\end{enumerate}


\end{enumerate}

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