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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Discussion 3 -- More on Limits}
\centerline{September 30, 2005}
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{\it Background}
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Recall that we have said that if $f(x)$ is a function defined
on some interval containing $x = c$, except possibly 
at $x=c$, then  $\lim_{x\to c} f(x) = L$ if we can make
the values $f(x)$ as close as we like to $L$ by taking
$x$ sufficiently close to $c$ (but not equal to $c$).
Today we want to gain more experience working with this
idea, considering limits graphically and numerically.
Similarly, we can define the ``one-sided'' limits
$\lim_{x\to c^+} f(x)$ and $\lim_{x\to c^-} f(x)$,
where we take $x$ sufficiently close to $c$ but only greater
than $c$ in the first case, and only less than $c$ in the second.
We also know that saying $f$ is {\it continuous at $x = c$}
should mean $\lim_{x\to c} f(x) = f(c)$.
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{\it Discussion Questions}
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A)  The following graph shows $y = f(x)$.  
\vglue 2.0 truein
\item{1)}  What are $\lim_{x\to 1^+} f(x)$ and 
$\lim_{x\to 1^-} f(x)$?
\item{2)}  Do you think $\lim_{x\to 1} f(x)$ exists?
Why or why not?
\item{3)}  Is $f$ continuous at $x = 1$?  Why or why not?
\item{4)}  What are $\lim_{x\to 3^+} f(x)$ and 
$\lim_{x\to 3^-} f(x)$?
\item{5)}  Do you think $\lim_{x\to 3} f(x)$ exists?
Why or why not?
\item{6)}  Is $f$ continuous at $x = 3$?  Why or why not?
\item{7)}  What are $\lim_{x\to 5^+} f(x)$ and 
$\lim_{x\to 5^-} f(x)$?
\item{8)}  Do you think $\lim_{x\to 5} f(x)$ exists?
Why or why not?
\item{9)}  Is $f$ continuous at $x = 5$?  Why or why not?
\item{10)}  In your own words, explain what must be true
about the one-sided limits at $x = c$ in order for 
$f$ to be continuous at $c$.  
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B) You will need a calculator for this question.  
Let $f(x) = {\sin(3x)\over x}$.
\item{1)}  Make a table of values for $f(x)$ at $x = -.1, -.01, -.001, -.0001, 
.0001, .001, .01, .1$.
\item{2)}  Make a conjecture (that is an ``educated guess'') 
about $\lim_{x\to 0} {\sin(3x)\over x}$ based on the values in your
table.  
\item{3)}  What does the graph $y = {\sin(3x)\over x}$ look like
for $x \in [-2,2]$.  Does this confirm your answer in 2?  Explain.
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{\it Please hand in your work to Prof. Ballantine at the end of class.}
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