Chapter 3.5, Problem 22E

To determine

To find: the end behavior of the given function.

Answer to Problem 22E

  $\begin{array}{l}f\left(x\right)\to \infty \text{ as }x\to \infty \\ f\left(x\right)\to \infty \text{ as }x\to -\infty \end{array}$

Explanation of Solution

Given information:

  $f\left(x\right)=x^{10}-x^9+5x^8$

Calculation:

Let $f\left(x\right)=y$ . Make a table for $x$ and y as follows:

$x$	$y$
$-10000$	$1.0001\times {10}^{40}$
$-1000$	$1.001\times {10}^{30}$
$-100$	$1.01\times {10}^{20}$
0	0
$100$	$9.905\times {10}^{19}$
$1000$	$9.99005\times {10}^{29}$
$10000$	$9.999001\times {10}^{39}$

The highest power of x is even and is positive, so as x approaches infinity the function approaches infinity and as x approaches negative infinity the graph approaches infinity. Thus, the end behavior is $y\to \infty \text{ as }x\to \infty$ and $y\to \infty \text{ as }x\to -\infty$ .