Chapter 11.4, Problem 36E

(a)

To determine

To find: The complete table given below and the limit as $x$ approaches to infinity for $f\left(x\right)=\frac{1}{x}-2$ .

$x$	${10}^0$	${10}^1$	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
$f\left(x\right)$

Answer to Problem 36E

The complete table is given below and the limit as $x$ approaches to infinity for $f\left(x\right)=\frac{1}{x}-2$ is equal to $-2$ .

$x$	${10}^0$	${10}^1$	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
$f\left(x\right)$	$-1$	$-1.9$	$-1.99$	$-1.999$	$-1.9999$	$-1.99999$	$-1.999999$

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{1}{x}-2$ and the table is:

$x$	${10}^0$	${10}^1$	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
$f\left(x\right)$

Calculation:

Substitute ${10}^0$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\frac{1}{x}-2\\ f\left({10}^0\right)=\frac{1}{{10}^0}-2\\ f\left({10}^0\right)=1-2\\ f\left({10}^0\right)=-1\end{array}$

So, the value of $f\left(x\right)$ for $x={10}^0$ is equal to $-1$ .

Substitute ${10}^1$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\frac{1}{x}-2\\ f\left({10}^1\right)=\frac{1}{{10}^1}-2\\ f\left({10}^1\right)=0.1-2\\ f\left({10}^1\right)=-1.9\end{array}$

So, the value of $f\left(x\right)$ for $x={10}^1$ is equal to $-1.9$ .

Substitute ${10}^2$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\frac{1}{x}-2\\ f\left({10}^2\right)=\frac{1}{{10}^2}-2\\ f\left({10}^2\right)=0.01-2\\ f\left({10}^2\right)=-1.99\end{array}$

So, the value of $f\left(x\right)$ for $x={10}^2$ is equal to $-1.99$ .

Substitute ${10}^3$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\frac{1}{x}-2\\ f\left({10}^3\right)=\frac{1}{{10}^3}-2\\ f\left({10}^3\right)=0.001-2\\ f\left({10}^3\right)=-1.999\end{array}$

So, the value of $f\left(x\right)$ for $x={10}^3$ is equal to $-1.999$ .

Substitute ${10}^4$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\frac{1}{x}-2\\ f\left({10}^4\right)=\frac{1}{{10}^4}-2\\ f\left({10}^4\right)=0.0001-2\\ f\left({10}^4\right)=-1.9999\end{array}$

So, the value of $f\left(x\right)$ for $x={10}^4$ is equal to $-1.9999$ .

Substitute ${10}^5$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\frac{1}{x}-2\\ f\left({10}^5\right)=\frac{1}{{10}^5}-2\\ f\left({10}^5\right)=0.00001-2\\ f\left({10}^5\right)=-1.99999\end{array}$

So, the value of $f\left(x\right)$ for $x={10}^5$ is equal to $-1.99999$ .

Substitute ${10}^6$ for $x$ in the function.

  $\begin{array}{c}f\left(x\right)=\frac{1}{x}-2\\ f\left({10}^6\right)=\frac{1}{{10}^6}-2\\ f\left({10}^6\right)=0.000001-2\\ f\left({10}^6\right)=-1.999999\end{array}$

So, the value of $f\left(x\right)$ for $x={10}^6$ is equal to $-1.999999$ .

So, the complete table is given below:

$x$	${10}^0$	${10}^1$	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
$f\left(x\right)$	$-1$	$-1.9$	$-1.99$	$-1.999$	$-1.9999$	$-1.99999$	$-1.999999$

As observed from the table the value of $f\left(x\right)$ tends to $-2$ approximately as $x$ tends to $\infty$ .

Therefore, the value of limit as $x$ approaches to infinity for $f\left(x\right)=\frac{1}{x}-2$ is equal to $-2$ .

(b)

To determine

To find: The graph of function $f\left(x\right)=\frac{1}{x}-2$ and the limit as $x$ tends to $\infty$ .

Answer to Problem 36E

The graph of function $f\left(x\right)=\frac{1}{x}-2$ is shown in figure(1) and the limit of $f\left(x\right)=\frac{1}{x}-2$ as $x$ tends to $\infty$ is equal to $-2$ .

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{1}{x}-2$ .

Calculation:

To graph a function $f\left(x\right)=\frac{1}{x}-2$ , 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 $y=\frac{1}{x}-2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{1}\boxed{÷}\boxed{\text{X}}\boxed{)}\boxed{-}\boxed{2}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(1/\text{X}\right)-2$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given quadratic equation in standard window as shown in Figure (1).

  

Figure (1)

As observed from the graph, the function $f\left(x\right)=\frac{1}{x}-2$ approaches to $-2$ as $x$ tends to $\infty$ .

Therefore, the limit of $f\left(x\right)=\frac{1}{x}-2$ as $x$ tends to $\infty$ is equal to $-2$ .

(c)

To determine

To find: The limit as $x$ approaches to infinity for $f\left(x\right)=\frac{1}{x}-2$ algebraically.

Answer to Problem 36E

The limit as $x$ approaches to infinity for $f\left(x\right)=\frac{1}{x}-2$ algebraically is equal to $-2$ .

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{1}{x}-2$ .

Calculation:

Simplify the limit.

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

Therefore, the limit as $x$ approaches to infinity for $f\left(x\right)=\frac{1}{x}-2$ algebraically is equal to $-2$ .