CPM Homework Help : APCALC Problem 3-178

What is the end-behavior function for each of the following functions?

Hint:

About end behavior:
End behavior describes the shape of the function if we ignore all vertical asymptotes and holes.

If the function has a horizontal asymptote, then that is its end behavior.
If a function has a slant asymptote, then that is it's end behavior.

If a function oscillates as $x \rightarrow \infty$ or $x \rightarrow -\infty$ , then it has no end behavior.

$f ( x ) = \large\frac { 2 x ^ { 2 } - 3 x + 1 } { x + 2 }$
Hint (a):
For rational functions, end behavior can often be found using polynomial division (and ignoring the remainder).

$g(x) = \frac { 1 } { x }+ \sin(x)$

Hint (b):
Use your calculator to sketch a graph of $g(x)$ . What does it look like as $x \rightarrow \infty$ and $x \rightarrow -\infty$ ?

$h ( x ) = \large\frac { \operatorname { sin } ( x ) } { x }$
Hint (c):
Examine.
$\lim\limits_{x\rightarrow \infty }g(x)\text{ and }\lim\limits_{x\rightarrow -\infty }g(x)$
Compare the numerator and denominator. Which has the highest power term as $x \rightarrow \infty$ ? (Remember that $y=\sin x$ never gets higher than $1$ or lower than $−1$ .) What does that say about asymptotes?