Chapter 2.3, Problem 60E

(a)

To determine

To find: The domain of the function $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$.

Answer to Problem 60E

The domain of the function $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$ is $\left(-\infty ,-1\right)\cup \left(0,\infty \right)$.

Explanation of Solution

Given information: The function is $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$.

Calculation:

The function $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$ is defined only if $1+\frac{1}{x}>0$.

Simplify the expression.

  $\begin{array}{l}1+\frac{1}{x}>0\\ \frac{x+1}{x}>0\end{array}$

This is possible only if $x<-1$ and $x>0$.

Therefore, the domain of the function $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$ is $\left(-\infty ,-1\right)\cup \left(0,\infty \right)$.

(b)

To determine

To graph: The function $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$.

Explanation of Solution

Given information: The function is $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$.

Graph:

To graph a function $y={\left(1+\frac{1}{x}\right)}^x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y={\left(1+\frac{1}{x}\right)}^x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{1}\boxed{+}\boxed{(}\boxed{1}\boxed{÷}\boxed{\text{X}}\boxed{)}\boxed{)}\boxed{^}\boxed{\text{X}}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(1+\left(1/\text{X}\right)\right)^\text{X}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

Interpretation: From the graph it can be observed that the function is continuous at every point in its domain except $x=-1$ and $x=0$.

(c)

To determine

To explain: The reason for discontinuity of $f\left(x\right)$ at $x=-1$ and $x=0$.

Answer to Problem 60E

The function $f\left(x\right)$ is discontinuous at $x=-1$ as the value is not in the domain of the function and the function is discontinuous at $x=0$ as the value of $f\left(x\right)$ at $x=0$ is not defined.

Explanation of Solution

Given information: The function is $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$.

Calculation:

As observed from the graph in part (b), the function $f\left(x\right)$ is not defined for $x=-1$ . So, the function is discontinuous at $x=-1$.

As observed from the graph in part (b), the value of function $f\left(x\right)$ is not defined at $x=0$.

The value of $f\left(0\right)$ is not defined. So, the function $f\left(x\right)$ is discontinuous at $x=0$.

Therefore, the function $f\left(x\right)$ is discontinuous at $x=-1$ and $x=0$.

(d)

To determine

To find: The point of removable discontinuity of $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$.

Answer to Problem 60E

The point of removable discontinuity of $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$ is $x=0$.

Explanation of Solution

Given information: The function is $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$.

Calculation:

As observed from the graph in part (b), the function $f\left(x\right)$ is not defined at $x=0$ but the right had limit at $x=0$ exists.

So, the point $x=0$ is the point of removable discontinuity as the extended function can be made continuous by setting the value of $f\left(0\right)$.

Therefore, the point of removable discontinuity of $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$ is $x=0$.

(e)

To determine

To find: The value of $\underset{x\to \infty }{\lim }f\left(x\right)$ for $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$ with the help of table and graph.

Answer to Problem 60E

The value of $\underset{x\to \infty }{\lim }f\left(x\right)$ is equal to $2.72$.

Explanation of Solution

Given information: The function is $f\left(x\right)={\left(1+\frac{1}{x}\right)}^x$.

Calculation:

As observed from the graph in part (b), the function $f\left(x\right)$ approaches to $2.72$ as $x$ approaches to infinity.

Now check the limit by table.

To make the table of function $y={\left(1+\frac{1}{x}\right)}^x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y={\left(1+\frac{1}{x}\right)}^x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{1}\boxed{+}\boxed{(}\boxed{1}\boxed{÷}\boxed{\text{X}}\boxed{)}\boxed{)}\boxed{^}\boxed{\text{X}}$.

First set the table setup, enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{WINDOW}}$ , then set to independent variable to ask.

Now draw the table, enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{GRAPH}}$ , and then enter the values of $\text{X}$ to $100$ , $200$ , $300$ , $400$ , $500$ , $1000$ and $2000$ . The table is shown below:

  

Figure (2)

As observed from table the value of $\text{Y}$ approaches to $2.72$ as $x$ approaches to $\infty$.

Therefore, the value of $\underset{x\to \infty }{\lim }f\left(x\right)$ is equal to $2.72$.