CPM Homework Help : CALC Problem 3-178

Find the end behavior function for 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 asymotote, 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 $x →∞$ or $x →−∞$ , then it has no end behavior.

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

$g ( x ) = \frac { 1 } { x } + \operatorname { sin } x$
Hint (b):
Use your calculator to sketch a graph of $g(x)$ . What does it look like as $x →∞$ and $x →−∞$ ?

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