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

3-178.

Find the end behavior function for 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 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.

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

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

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

Use your calculator to sketch a graph of $g(x)$. What does it look like as $x →∞$ and $x →−∞$?

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

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?