Chapter 12.4, Problem 70E

a.

To determine

Use the graph to estimate $\underset{x\to \infty }{\lim }f\left(x\right)$ .

Answer to Problem 70E

  $\boxed{y=0}$ and $\boxed{y=2}$

Explanation of Solution

Given information:

Use the graph to estimate $\underset{x\to \infty }{\lim }f\left(x\right)$

  

Calculation:

Consider the following graph

  

Look at the above graph of the function $f(x)$ . We see that $f(x)$ gets close to $0$ as $x$ increases or decreases without bound. So,

  $\underset{x\to \infty }{\lim }f\left(x\right)=0$

Hence, $y=0$ is a horizontal asymptote of the graph of the function $f$ .

Consider the following graph

  

Look at the above graph of the function $f(x)$ . We see that $f(x)$ gets close to $2$ as $x$ increases or decreases without bound. So,

  $\underset{x\to \infty }{\lim }f\left(x\right)=2$

Hence, $y=2$ is a horizontal asymptote of the graph of the function $f$ .

b.

To determine

Use the graph to estimate $\underset{x\to -\infty }{\lim }f\left(x\right)$

Answer to Problem 70E

  $\boxed{y=0}$ and $\boxed{y=2}$

Explanation of Solution

Given information:

Use the graph to estimate $\underset{x\to -\infty }{\lim }f\left(x\right)$

  

Calculation:

Consider the following graph

  

Look at the above graph of the function $f(x)$ . We see that $f(x)$ gets close to $0$ as $x$ increases or decreases without bound. So,

  $\underset{x\to -\infty }{\lim }f\left(x\right)=0$

Hence, $y=0$ is a horizontal asymptote of the graph of the function $f$ .

Consider the following graph

  

Look at the above graph of the function $f(x)$ . We see that $f(x)$ gets close to $2$ as $x$ increases or decreases without bound. So,

  $\underset{x\to -\infty }{\lim }f\left(x\right)=2$

Hence, $y=2$ is a horizontal asymptote of the graph of the function $f$ .

c.

To determine

Use the graph to estimate the horizontal asymptote of the graph of $f$ .

Answer to Problem 70E

  $\boxed{S}$

Explanation of Solution

Given information:

Use the graph to estimate the horizontal asymptote of the graph of $f$ .

  

Calculation:

Hence $y=2$ is a horizontal asymptote of the graph of the function $f$ . $\boxed{S}$