Chapter 2.2, Problem 45E

(a)

To determine

To find:A simple function as a right end behavior model for the function $y=e^x-2x$ .

Answer to Problem 45E

A simple function as a right end behavior model for the function $y=e^x-2x$ is $e^x$ .

Explanation of Solution

Given information:

The function is $y=e^x-2x$ .

Calculation:

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

As $x\to \infty$ , then the function $e^x$ increases 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.

Verify:

  $\begin{array}{c}\underset{x\to \infty }{\lim }\frac{e^x-2x}{e^x}=\underset{x\to \infty }{\lim }\left(1-\frac{2x}{e^x}\right)\\ =1-0\\ =1\end{array}$

Therefore, a simple function as a right end behavior model for the function $y=e^x-2x$ is $e^x$ .

(b)

To determine

To find:A simple function as a left end behavior model for the function $y=e^x-2x$ .

Answer to Problem 45E

A simple function as a left end behavior model for the function $y=e^x-2x$ is $-2x$ .

Explanation of Solution

Given information:

The function is $y=e^x-2x$ .

Calculation:

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

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

The function $f\left(x\right)=-2x$ will be the right-end behavior model.

Verify:

  $\begin{array}{c}\underset{x\to \infty }{\lim }\frac{e^x-2x}{-2x}=\underset{x\to \infty }{\lim }\left(1-\frac{e^x}{2x}\right)\\ =1-0\\ =1\end{array}$

Therefore, a simple function as a left end behavior model for the function $y=e^x-2x$ is $-2x$ .