\def\wsn{4}
\input worksheet.tex

\itemitem{1.\hskip12pt a)}  For each of the limits, graph the
expression given, then use your graph to evaluate the limit.
$$
\lim_{x\to-1^+}{x^2+x\over|x^2+x|}\hskip1in
\lim_{x\to1}{(1-|x-1|)}
$$
\itemitem{b)}  For each of the limits below, use your calculator to
evaluate the expression for at least 4 different values of $x$.  Use
this data to guess the limit.
$$
\lim_{x\to0^+}{x-\sin x\over x}\hskip.75in
\lim_{x\to0}{\sin^2(2x)\over x\sin x}\hskip.75in
\lim_{x\to0}{x\over\tan x}
$$

\item{}  Note: Four of these five problems have appeared on
Dr. Davis's previous exams.

\bigskip
\item{2.}  A function is said to be {\it continuous at the point}
$x_0$ if 
$$
\eqalign{
\hbox{i. }&f(x_0) \quad\hbox{is defined;}\cr
\hbox{ii. }&\lim_{x\to x_0}f(x) \quad\hbox{exists;}\cr
\hbox{iii. }&\hbox{and }\lim_{x\to x_0}f(x)=f(x_0).\cr}
$$

\itemitem{a)}  Sketch a graph of a discontinuous function for each of
the following: 
\itemitem{} - condition i. holds, but condition ii. does not
\itemitem{} - condition ii. holds, but condition i. does not
\itemitem{} - conditions i. and ii. both hold, but condition iii. does
not

\medskip
\itemitem{b)}  Classify your examples as removable discontinuities,
jump discontinuities, or asymptotes.

\medskip
\itemitem{c)}  Could you have drawn examples which would have been
classified differently?

\bigskip
\item{3.}  Determine whether the following functions are continuous at
the points given.  At discontinuous points, indicate which of the
conditions from Problem 2 do not hold.
\medskip
\settabs 3\columns
\+$\qquad\pt a x_0=-2,1$&$\qquad\pt b x_0=-3,1$&$\qquad\pt c x_0=-1,0,2$\cr
\medskip
\+\qquad\epsfxsize=1.8in\epsfbox{/home/oehrtman/m210/graph1.eps}
&\qquad\epsfxsize=1.8in\epsfbox{/home/oehrtman/m210/graph2.eps}
&\qquad\epsfxsize=1.8in\epsfbox{/home/oehrtman/m210/graph3.eps}\cr

\bigskip
\item{4.}  Graph the following functions:
$$
f(x)=\cases{x^2+5,&$x<-1$;\cr6x,&$x=-1$;\cr27x+33,&$x>-1$.\cr}
\hskip1in
g(x)=\cases{\theta\sin{1\over\theta},&$\theta\neq0$;\cr\cr0,&$\theta=0$.}
$$
Determine whether $f(x)$ is continuous at $x_0=-1$ and whether $g(x)$
is continuous at $x_0=0$.  Fully justify your answers using the three
parts of the definition of continuity given in Problem 2.


\eject
\item{5.}  Suppose $f$, $g$, and $h$ are functions and that 
$$
g(x)\leq f(x)\leq h(x) \qquad \forall x\in{\bf R}.
$$
Furthermore, suppose that
$$
\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L.
$$
What is $\lim_{x\to a}f(x)$?  Why?

\item{} Hint: Draw a picture.

\bigskip
\item{6.}  Evaluate the following limits algebraically  (you may need
to be clever for some of them.)
$$
\displaylines{
\pt a \lim_{x\to3}{\sqrt x-\sqrt3\over x-3}
\qquad
\pt b \lim_{x\to-2}{x^2+5x+6\over x^2+x-2}
\qquad
\pt c \lim_{x\to0}{\sqrt{4+h}-2\over h}
\qquad
\pt d \lim_{x\to8}{\root 3 \of{x}-2\over x-8}\cr\cr
\pt e \lim_{x\to3}{1/x-1/3\over x-3}\hskip1in
\pt f \lim_{x\to0}(1+x)^{1/x}\cr}
$$


\bigskip
\item{7.}  Given below is a {\it one-parameter family of functions}.
That is, it is a collection of many functions $f_t$ - one for each
value of the parameter $t$.  Determine which values of this parameter
give a continuous function.
$$
f_t(x)=\cases{t\sin x,&$x\leq{\pi\over2}$;\cr\cr
(1-t)x+t^2,&$x>{\pi\over2}$.\cr} 
$$
Graph $f_t$ for every value of $t$ for which $f_t$ is continuous.



\bye