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\begin{document}
\leftline{MATH 131, section 1 -- Calculus for Physical and Life Sciences}
\leftline{First Limit Examples}
\leftline{September 24, 2007}
\vskip 20pt
\noindent
{\it Example 1.}  Investigate the behavior of $f(x) = \cos(x)$ for $x$ near
$a = \frac{\pi}{3} \doteq 1.0472$ radians.  Can we ``spot the trend?''
\[
\begin{array}{cccccccccc}
x & \vline & .9 & 1.0 & 1.04 & 1.047 & 1.0471 & 1.048 & 1.05 & 1.1 \\ \hline
\cos(x) & \vline & .62161 & .54030  & .50622 & .50017 & .50008 & .49930 & .49757 & .45360 
\end{array}
\]
(Note: all values are rounded to 5 decimal places.)
\vskip 10pt
\noindent
Guess:  $\displaystyle{\lim_{x\to \frac{\pi}{3}} \cos(x)} = \underline{\qquad\qquad\qquad}$.
\vskip 20pt
\noindent
{\it Example 2.}
Investigate the behavior of $h(x) = \frac{x^2 - 4x + 3}{x - 1}$
for $x$ near $a = 1$.  What is the trend here?
\[
\begin{array}{cccccccccc}
x & \vline & 0.0 & 0.5 & 0.9 & 0.99 & 1.01 & 1.1 & 1.5 & 2.0 \\ \hline
\frac{x^2-4x+3}{x-1} & \vline & -3.0 & -2.5  & -2.1 & -2.01 & -1.99 & -1.9 & -1.5 & -1
\end{array}
\]
\vskip 10pt
\noindent
Guess:  $\displaystyle{\lim_{x\to 1} \frac{x^2-4x+3}{x-1}} = \underline{\qquad\qquad\qquad}$.
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\noindent
{\it Example 3.}  Investigate the behavior of $g(x) = x\sin\left(\frac{1}{x}\right)$
for $x$ near $a = 0$.  What is the trend here?
\[
\begin{array}{cccccccccc}
x & \vline & -1.0 & -.1 & -.01 & -.001 & .001 & .01 & .1 & 1.0 \\ \hline
x\sin\left(\frac{1}{x}\right) & \vline & .84147 & -.05440  & -.00506 & .00083 & .00083 & -.00506 & -.05440 & .84147 
\end{array}
\]
\vskip 10pt
\noindent
Guess:  $\displaystyle{\lim_{x\to 0} x\sin\left(\frac{1}{x}\right)} = \underline{\qquad\qquad\qquad}$.
\vskip 10 pt
\noindent
Here the graph explains the apparent frequent oscillations from positive to negative
values and back (turn over for graph).

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\includegraphics[width=2.5truein,height=2.5truein]{Limit.ps} 
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\centerline{The graph $y = x \sin\left(\frac{1}{x}\right)$ for $x$ near $0$.}
\end{document}
