Chapter 11.4, Problem 72E

a.

To determine

To estimate: $\underset{x\to \infty }{\lim }f\left(x\right)$ for the given two graphs.

Answer to Problem 72E

For graph (i) the value of the limits is $\underset{x\to \infty }{\lim }f\left(x\right)=0$

For graph (ii) the value of the limits is $\underset{x\to \infty }{\lim }f\left(x\right)=2$

Explanation of Solution

Given:

The graphs are:-

  

Concept used:

The value of $f\left(x\right)$ is equal to the value of $y$ .

Calculation:

The graph (i) is

  

From the graph it can be seen that when the value of x tends to $\infty$ then the value of y approaches to 0.

So,

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

The graph (ii) is

  

From the graph it can be seen that when the value of x tends to $\infty$ , then the value of y approaches to 2. So the value of f approaches to 2.

So,

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

b.

To determine

To estimate: $\underset{x\to -\infty }{\lim }f\left(x\right)$ for the given two graphs.

Answer to Problem 72E

For graph (i) the value of the limits is $\underset{x\to -\infty }{\lim }f\left(x\right)=0$

For graph (ii) the value of the limits is $\underset{x\to -\infty }{\lim }f\left(x\right)=2$

Explanation of Solution

Given:

The graphs are

  

Concept used:

The value of $f\left(x\right)$ is equal to $y$ .

Calculation:

The graph (i) is

  

From the graph it can be seen that when the value of x tends to $-\infty$ , then the value of y approaches to 0.

So,

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

The graph (ii) is

  

From the graph it can be seen that when the value of x tends to $-\infty$ , then the value of y approaches to 2.

So,

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

c.

To determine

To find: The horizontal asymptote of the graph of f .

Answer to Problem 72E

For graph (i) the horizontal asymptote is $y=0$ .

For graph (ii) the horizontal asymptote is $y=2$ .

Explanation of Solution

Given:

The graphs are

  

Concept used:

Horizontal asymptote: in the graph of f , $y=c$ is called horizontal asymptote if $\underset{x\to \infty }{\lim }f\left(x\right)=c$ or $\underset{x\to -\infty }{\lim }f\left(x\right)=c$ .

Calculation:

The graph (i) is

  

From the graph it can be seen that when the value of x tends to $\pm \infty$ then the value of y approaches to 0.

So the value of f approaches to 0.

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

Hence according to the definition, the horizontal asymptote of the graph (i) is $y=0$ .

The graph (ii) is

  

From the graph it can be seen that when the value of x tends to $\pm \infty$ the value of y approaches to 2. So the value of f approaches to 2.

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

Hence according to the definition, the horizontal asymptote of the graph (ii) is $y=2$ .