\magnification 2000
\parindent 0pt
\parskip 12pt
\pageno=1
\nopagenumbers
\hsize 7.5truein
\hoffset -0.35truein
%%\voffset -0.1truein
\vsize 9.5truein
\def\u{\vskip -10pt}
\def\v{\vskip 5pt}
\def\s{\vskip -5pt}
\def\r{\vskip -4pt}
\def\w{\vskip 7pt}




$lim_{x \rightarrow 3}~ {x^2 - 1 \over x + 3}$ 



\vfill



$lim_{x \rightarrow 3}~ {x^2 - 1 \over x - 3}$ 



\vfill



$lim_{x \rightarrow 3}~ {(x^2 - 1)(x - 3)\over x - 3}$ 



\vfill





$lim_{x \rightarrow 3}~ { x - 3 \over x^2 - 1 }$ 



\vfill





$lim_{x \rightarrow 3}~ {(x-4)^2 \over x^5(x-8)^9(x - 3)^3}$ 



\vfill





$lim_{x \rightarrow 3}~ {(x-4)^2(x - 3) \over x^5(x-8)^9(x - 3)^3}$ 



\vskip 5pt
\hrule
\vskip -5pt

Challenge example: ~~ $g(x) = x sin{1 \over x} $
\w\w
\centerline{$-|x| \leq x sin{1 \over x} \leq |x|$ \hskip 1in}
\w\w
 \centerline{$lim_{x \rightarrow 0} \big(-|x|\big) = 0$,  $lim_{x 
\rightarrow 0}
\big(|x|\big) = 0$.}
Hence ,  $lim_{x \rightarrow 0} \big(x sin{1 \over x}\big)  = 0$


\eject


Standard example:  \hfil \break
Suppose $f(x) = \sqrt{x}$.  Find $lim_{h \rightarrow 0} 
{f(x + h) - f(x) \over h}$ where $x > 0$

\eject
Suppose $c \in {\cal R}$ and suppose 
$lim_{x \rightarrow a} f(x)$ and $lim_{x \rightarrow a} g(x)$ exist.
Then
\s

$lim_{x \rightarrow a} [f(x) + g(x)] $ = $lim_{x \rightarrow a} f(x)$ + $lim_{x \rightarrow a} g(x)$ 

$lim_{x \rightarrow a} [cf(x)] = $ $c ~lim_{x \rightarrow a} f(x)$ 

$lim_{x \rightarrow a} [f(x)g(x)] = $ $lim_{x \rightarrow a} f(x)$ $lim_{x \rightarrow a} g(x)$ 

$lim_{x \rightarrow a} {f(x) \over g(x)} = $ 
${lim_{x \rightarrow a} f(x) \over lim_{x \rightarrow a} g(x)}$ if 
$lim_{x \rightarrow a} g(x) \not= 0$  

\vskip 5pt
\hrule
%%Defn:  $f$ is continuous at $a$ \hfil \break
%%iff $lim_{x \rightarrow a} f(x) = f(lim_{x \rightarrow a} x) = $ 

  Defn:  $f$ is continuous at $a$ if $lim_{x \rightarrow a} f(x) = f(a)$

\centerline{(i.e., if $lim_{x \rightarrow a} f(x) = f(lim_{x \rightarrow 
a} x) $}
\s
In other words, $f$ is continuous at $a$ if \hfil\break
1.)  $f(a)$ exists,\hfil\break
2.) $lim_{x \rightarrow a} f(x)$ exists, and\hfil\break
3.) $lim_{x \rightarrow a} f(x) = 
f(a)$


Defn:  $f$ is continuous is $f$ is continuous at $a$ for every $a$ in the 
domain of $f$.


Examples:

\vfill

Ex:  Polynomial, rational, root, trigonometric, inverse trigonometric, 
exponential,
logarithmic functions are continuous functions.

%%\eject


If $f$, $g$ continuous at $a$, $c \in {\cal R}$, then $f + g$, $fg$, $cf$, 
$f/g$ (if $g(a) \not=  0$)
are continuous at $a$.



If $g$ continuous at $a$ and $f$ continuous at $g(a)$, then $f \circ g$ 
continuous at $a$.


Ex:  $lim_{x \rightarrow 0}~ {x^2 - e^{x^3} \over cos(x)} =$

\vfill

%%If $f$ is continuous implies 
%% \centerline{$lim_{x \rightarrow a} f(g(x)) = f(lim_{x \rightarrow 
%% a}g(x))$ }


Ex:  $lim_{x \rightarrow 9}~ e^{\sqrt{x} } - 2\sqrt{x} + 4 =$ 
\vfill

Ex:  $lim_{x \rightarrow 0}~ cos(sin(x)) =$
\vfill

Ex:  $lim_{x \rightarrow 0}~ cos({sin(x) \over x}) =$
\vfill

Ex:  $lim_{h \rightarrow 0}~ {(h)tan(x)csc(h)} =$

\vfill






\end