Chapter 2.1, Problem 19E

(a)

To determine

To find: Complete the table given below for $f\left(x\right)=\frac{{10}^x-1}{x}$ and also estimate the value of $\underset{x\to 0}{\lim }f\left(x\right)$ :

$x$	$-0.1$	$-0.01$	$-0.001$	$-0.0001$
$f\left(x\right)$	$?$	$?$	$?$	$?$

Answer to Problem 19E

The value of the $\underset{x\to 0}{\lim }f\left(x\right)$ is $2.31$ and the complete table is given below:

$x$	$-0.1$	$-0.01$	$-0.001$	$-0.0001$
$f\left(x\right)$	$2.0567$	$2.2762$	$2.2999$	$2.3023$

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{{10}^x-1}{x}$ .

Calculation:

Substitute $-0.1$ for $x$ in the given function.

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

To find the value of $-\frac{{10}^{-0.1}-1}{0.1}$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(-)}\boxed{(}\boxed{10}\boxed{^}\boxed{(-)}\boxed{0.1}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{0.1}$

The value of $-\frac{{10}^{-0.1}-1}{0.1}$ using a calculator is approximately $2.0567$ .

So, the value of $f\left(x\right)$ at $x=-0.1$ is $2.0567$ .

Substitute $-0.01$ for $x$ in the given function.

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

To find the value of $-\frac{{10}^{-0.01}-1}{0.01}$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(-)}\boxed{(}\boxed{10}\boxed{^}\boxed{(-)}\boxed{0.01}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{0.01}$

The value of $-\frac{{10}^{-0.01}-1}{0.01}$ using a calculator is approximately $2.2762$ .

So, the value of $f\left(x\right)$ at $x=-0.01$ is $2.2762$ .

Substitute $-0.001$ for $x$ in the given function.

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

To find the value of $-\frac{{10}^{-0.001}-1}{0.001}$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(-)}\boxed{(}\boxed{10}\boxed{^}\boxed{(-)}\boxed{0.001}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{0.001}$

The value of $-\frac{{10}^{-0.001}-1}{0.001}$ using a calculator is approximately $2.2999$ .

So, the value of $f\left(x\right)$ at $x=-0.001$ is $2.2999$ .

Substitute $-0.0001$ for $x$ in the given function.

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

To find the value of $-\frac{{10}^{-0.0001}-1}{0.0001}$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(-)}\boxed{(}\boxed{10}\boxed{^}\boxed{(-)}\boxed{0.0001}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{0.0001}$

The value of $-\frac{{10}^{-0.0001}-1}{0.0001}$ using a calculator is approximately $2.3023$ .

So, the value of $f\left(x\right)$ at $x=-0.0001$ is $2.3023$ .

Therefore, the complete table is given by:

$x$	$-0.1$	$-0.01$	$-0.001$	$-0.0001$
$f\left(x\right)$	$2.0567$	$2.2762$	$2.2999$	$2.3023$

The value of function tends to $2.31$ as $x$ tends to $0$ .

Therefore, The value of the $\underset{x\to 0}{\lim }f\left(x\right)$ is $2.31$ .

(b)

To determine

To find: Complete the table given below for $f\left(x\right)=\frac{{10}^x-1}{x}$ and also estimate the value of $\underset{x\to 0}{\lim }f\left(x\right)$ :

$x$	$0.1$	$0.01$	$0.001$	$0.0001$
$f\left(x\right)$	$?$	$?$	$?$	$?$

Answer to Problem 19E

The value of the $\underset{x\to 0}{\lim }f\left(x\right)$ is $2.31$ and the complete table is given below:

$x$	$0.1$	$0.01$	$0.001$	$0.0001$
$f\left(x\right)$	$2.5892$	$2.3292$	$2.3052$	$2.3028$

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{{10}^x-1}{x}$ .

Calculation:

Substitute $0.1$ for $x$ in the given function.

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

To find the value of $\frac{{10}^{0.1}-1}{0.1}$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(}\boxed{10}\boxed{^}\boxed{0.1}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{0.1}$

The value of $\frac{{10}^{0.1}-1}{0.1}$ using a calculator is approximately $2.5892$ .

So, the value of $f\left(x\right)$ at $x=0.1$ is $2.5892$ .

Substitute $0.01$ for $x$ in the given function.

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

To find the value of $\frac{{10}^{0.01}-1}{0.01}$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(}\boxed{10}\boxed{^}\boxed{0.01}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{0.01}$

The value of $\frac{{10}^{0.01}-1}{0.01}$ using a calculator is approximately $2.3292$ .

So, the value of $f\left(x\right)$ at $x=0.01$ is $2.3292$ .

Substitute $0.001$ for $x$ in the given function.

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

To find the value of $\frac{{10}^{0.001}-1}{0.001}$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(}\boxed{10}\boxed{^}\boxed{0.001}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{0.001}$

The value of $\frac{{10}^{0.001}-1}{0.001}$ using a calculator is approximately $2.3052$ .

So, the value of $f\left(x\right)$ at $x=0.001$ is $2.3052$ .

Substitute $0.0001$ for $x$ in the given function.

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

To find the value of $\frac{{10}^{0.0001}-1}{0.0001}$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(}\boxed{10}\boxed{^}\boxed{0.0001}\boxed{-}\boxed{1}\boxed{)}\boxed{÷}\boxed{0.0001}$

The value of $\frac{{10}^{0.0001}-1}{0.0001}$ using a calculator is approximately $2.3028$ .

So, the value of $f\left(x\right)$ at $x=0.0001$ is $2.3028$ .

Therefore, the complete table is given by:

$x$	$0.1$	$0.01$	$0.001$	$0.0001$
$f\left(x\right)$	$2.5892$	$2.3292$	$2.3052$	$2.3028$

The value of function tends to $2.30$ as $x$ tends to $0$ .

Therefore, The value of the $\underset{x\to 0}{\lim }f\left(x\right)$ is $2.31$ .