Chapter 2, Problem 37RE

(a)

To determine

To find: The right end behavior model for the function $f\left(x\right)=x+e^x$ .

Answer to Problem 37RE

The right end behavior model is $e^x$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=x+e^x$ .

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.

  $f\left(x\right)=x+e^x$

As $x\to \infty$ , then the function $e^x$ increases exponentially and it tends much away from the x -axis. So, the term $e^x$ dominates on the right.

The function $f\left(x\right)=e^x$ will be the right-end behavior model for the given function.

Therefore, the right end behavior model for the function is $e^x$ .

(b)

To determine

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

Answer to Problem 37RE

The left end behavior model for the function is $x$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=x+e^x$ .

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.

  $f\left(x\right)=x+e^x$

As $x\to -\infty$ , then the function $x$ approaches to infinity. So, the term $x$ dominates on the left.

Therefore, the left end behavior model is $x$ .