### Home > APCALC > Chapter 3 > Lesson 3.4.3 > Problem3-178

3-178.

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

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.

1. $f ( x ) = \large\frac { 2 x ^ { 2 } - 3 x + 1 } { x + 2 }$

For rational functions, end behavior can often be found using polynomial division (and ignoring the remainder).

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

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

1. $h ( x ) = \large\frac { \operatorname { sin } ( x ) } { x }$

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?