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Section 2.3


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{$lim_{x \rightarrow 0} \big(-|x|\big) = 0$,  $lim_{x \rightarrow 0} \big(|x|\big) = 0$.}  
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Hence ,  $lim_{x \rightarrow 0} \big(x sin{1 \over x}\big)  = 0$   

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Section 2.5

  Defn:  $f$ is continuous at $a$ if $lim_{x \rightarrow a} f(x) = f(a)$
\s
(i.e., if $lim_{x \rightarrow a} f(x) = f(lim_{x \rightarrow a} x) $
\s
Examples:

\vskip 1.8in
\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.

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


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

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Section 3.3: 
To find vertical asymptotes,\hfil \break
 find all $a \in {\cal R}$ such that \hfil \break
$lim_{x \rightarrow a^-} f(x) = \pm \infty $ and/or
$lim_{x \rightarrow a^+} f(x) = \pm \infty $

Ex:  $f(x) = {1 \over (x+2)(x -3)^2}$

\vskip 1.1in

Section 2.6: \hfil \break
  Horizontal asymptotes/limits at infinity

\s

To find horizontal asymptotes: \hfil\break
 calculate $lim_{x \rightarrow +\infty} f(x)$ and  
$lim_{x \rightarrow -\infty} f(x)$ 

\s

IF $lim_{x \rightarrow +\infty} f(x) = L$ where $L$ is a finite real number, then $y = L$ is a 
horizontal asymptote.  
\s

IF $lim_{x \rightarrow -\infty} f(x) = K$ where $K$ is a finite real number, then $y = K$ is a 
horizontal asymptote.  

\vfill
\eject

Ex: $f(x) = {2x^3 - x^2 + 1 \over 8x^3 + x + 3}$ 

$lim_{x \rightarrow +\infty} {2x^3 - x^2 + 1 \over 8x^3 + x + 3} = $
\vfill

Similarly, $lim_{x \rightarrow -\infty} {2x^3 - x^2 + 1 \over 8x^3 + x + 3} = $

Horizontal asymptote(s):
\vskip 1cm
\eject

Ex: $f(x) = { x^2 + 1 \over 2x^5 + x^2 - 3}$   

$lim_{x \rightarrow +\infty} { x^2 + 1 \over 2x^5 + x^2 - 3} = $

\vfill

Similarly, $lim_{x \rightarrow -\infty} { x^2 + 1 \over 2x^5 + x^2 - 3} = $

Horizontal asymptote(s):
\v
\hrule

Ex: $f(x) = { 2x^5 + x^2 - 3 \over x^2 + 1 }$  

$lim_{x \rightarrow +\infty} { 2x^5 + x^2 - 3 \over x^2 + 1 } = $
\vfill

Also, $lim_{x \rightarrow -\infty} { 2x^5 + x^2 - 3 \over x^2 + 1 }= $

Horizontal asymptote(s):
\v
\eject

Ex: $f(x) = { 2x \over \sqrt {x^2 + 1} }  $  

$lim_{x \rightarrow +\infty} { 2x \over \sqrt{x^2 + 1} } = $
\vfill

$lim_{x \rightarrow -\infty} { 2x \over \sqrt{x^2 + 1} }= $
\vfill
\v\v\v\v\v\v\v\v\v\v\v
Horizontal asymptote(s):
\v
\eject

Ex:  $f(x) = x^2 - x^3$

$lim_{x \rightarrow +\infty} x^2 - x^3 = $
\vfill
\vfill\vfill
$lim_{x \rightarrow -\infty} x^2 - x^3 = $
\vfill\vfill
\vfill
Horizontal asymptote(s):
\v
\hrule

Ex:  $f(x) = x^{2 \over 3} - x$
 
$lim_{x \rightarrow +\infty} x^{2 \over 3} - x =$
\vfill
\vfill\vfill
$lim_{x \rightarrow -\infty} x^{2 \over 3} - x =$
\vfill
\vfill
\vfill
Horizontal asymptote(s):

\v

\end