Chapter 1, Problem 38RE

(a)

To determine

To find: The right end behavior model.

Answer to Problem 38RE

  $g(x)=\ln \left|x\right|$ is a right end behavior model.

Explanation of Solution

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

Calculation:

The general direction of $f(x)$ will be dictated by $\ln x$ in both directions because $\sin x$ simply oscillates between -1 and 1 when moving in either direction. Therefore, when moving to the right, the direction is $\ln x$ (the abs. value is not required in that direction).

According to Eq. (1), it follows that $g(x)=\ln \left|x\right|$ indeed is a right-end behavior model for the function $f(x)=\ln \left|x\right|+\sin x$ .

(b)

To determine

To find: The left end behavior model.

Answer to Problem 38RE

  $g(x)=\ln \left|x\right|$ is a left end behavior model.

Explanation of Solution

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

Calculation:

The general direction of $f(x)$ will be dictated by $\ln x$ in both directions because $\sin x$ simply oscillates between -1 and 1 when moving in either direction. Therefore, it is $\ln x$ while moving to the left.

Therefore, the required $g(x)=\ln \left|x\right|$ is a left end behavior model.