Chapter 2, Problem 38RE

(a)

To determine

To find: The right end behavior model for the given function.

Answer to Problem 38RE

The right end behavior is $\ln \left|x\right|$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=\ln \left|x\right|+\sin x$ .

Calculation:

A function $f\left(x\right)$ is said to be right end behavior model for the function $g\left(x\right)$ if $\underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}=1$ .

Consider the function.

  $f\left(x\right)=\ln \left|x\right|+\sin x$

In given function the value of $\sin x$ lie between $-1$ to $1$ . As $x\to \infty$ the function is decided by the value of $\ln \left|x\right|$ .

Therefore, the right end behavior is $\ln \left|x\right|$ .

(b)

To determine

To find: The left end behavior model for the function.

Answer to Problem 38RE

The left end behavior model for the function is $\ln \left|x\right|$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=\ln \left|x\right|+\sin x$ .

Calculation:

A function $f\left(x\right)$ is said to be left end behavior model for the function $g\left(x\right)$ if $\underset{x\to -\infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}=1$ .

Consider the function.

  $f\left(x\right)=\ln \left|x\right|+\sin x$

In given function the value of $\sin x$ lie between $-1$ to $1$ . As $x\to -\infty$ the function is decided by the value of $\ln \left|x\right|$ .