Chapter 2.1, Problem 20E

(a)

To determine

To find: Complete the table given below for $f\left(x\right)=x\sin \left(\ln \left|x\right|\right)$ 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 20E

The value of the $\underset{x\to 0}{\lim }f\left(x\right)$ doesn’t exist and the complete table is given below:

$x$	$-0.1$	$-0.01$	$-0.001$	$-0.0001$
$f\left(x\right)$	$0.0743$	$-0.0099$	$0.0005$	$0.00002$

Explanation of Solution

Given information:

The function is $f\left(x\right)=x\sin \left(\ln \left|x\right|\right)$ .

Calculation:

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

  $\begin{array}{c}f\left(x\right)=x\sin \left(\ln \left|x\right|\right)\\ f\left(-0.1\right)=-0.1\times \sin \left(\ln \left|-0.1\right|\right)\\ f\left(-0.1\right)=-0.1\times \sin \left(\ln \left(0.1\right)\right)\end{array}$

To find the value of $-0.1\times \sin \left(\ln \left(0.1\right)\right)$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(-)}\boxed{0.1}\boxed{\times }\boxed{\text{SIN}}\boxed{\text{LN}}\boxed{0.1}\boxed{)}\boxed{)}$

The value of $-0.1\times \sin \left(\ln \left(0.1\right)\right)$ using a calculator is approximately $0.0743$ .

So, the value of $f\left(x\right)$ at $x=-0.1$ is $0.0743$ .

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

  $\begin{array}{c}f\left(x\right)=x\sin \left(\ln \left|x\right|\right)\\ f\left(-0.01\right)=-0.01\times \sin \left(\ln \left|-0.01\right|\right)\\ f\left(-0.01\right)=-0.01\times \sin \left(\ln \left(0.01\right)\right)\end{array}$

To find the value of $-0.01\times \sin \left(\ln \left(0.01\right)\right)$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(-)}\boxed{0.01}\boxed{\times }\boxed{\text{SIN}}\boxed{\text{LN}}\boxed{0.01}\boxed{)}\boxed{)}$

The value of $-0.01\times \sin \left(\ln \left(0.01\right)\right)$ using a calculator is approximately $-0.0099$ .

So, the value of $f\left(x\right)$ at $x=-0.01$ is $-0.0099$ .

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

  $\begin{array}{c}f\left(x\right)=x\sin \left(\ln \left|x\right|\right)\\ f\left(-0.001\right)=-0.001\times \sin \left(\ln \left|-0.001\right|\right)\\ f\left(-0.001\right)=-0.001\times \sin \left(\ln \left(0.001\right)\right)\end{array}$

To find the value of $-0.001\times \sin \left(\ln \left(0.001\right)\right)$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(-)}\boxed{0.001}\boxed{\times }\boxed{\text{SIN}}\boxed{\text{LN}}\boxed{0.001}\boxed{)}\boxed{)}$

The value of $-0.001\times \sin \left(\ln \left(0.001\right)\right)$ using a calculator is approximately $0.0005$ .

So, the value of $f\left(x\right)$ at $x=-0.001$ is $0.0005$ .

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

  $\begin{array}{c}f\left(x\right)=x\sin \left(\ln \left|x\right|\right)\\ f\left(-0.0001\right)=-0.0001\times \sin \left(\ln \left|-0.0001\right|\right)\\ f\left(-0.0001\right)=-0.0001\times \sin \left(\ln \left(0.0001\right)\right)\end{array}$

To find the value of $-0.0001\times \sin \left(\ln \left(0.0001\right)\right)$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{(-)}\boxed{0.0001}\boxed{\times }\boxed{\text{SIN}}\boxed{\text{LN}}\boxed{0.0001}\boxed{)}\boxed{)}$

The value of $-0.0001\times \sin \left(\ln \left(0.0001\right)\right)$ using a calculator is approximately $0.00002$ .

So, the value of $f\left(x\right)$ at $x=-0.0001$ is $0.00002$ .

Therefore, the complete table is given by:

$x$	$-0.1$	$-0.01$	$-0.001$	$-0.0001$
$f\left(x\right)$	$0.0743$	$-0.0099$	$0.0005$	$0.00002$

The value of the function is not possible to find when $x$ tends to $0$ as the values of function are oscillates with decrease in $x$ .

Therefore, The value of the $\underset{x\to 0}{\lim }f\left(x\right)$ doesn’t exist.

(b)

To determine

To find: Complete the table given below for $f\left(x\right)=x\sin \left(\ln \left|x\right|\right)$ 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 20E

The value of the $\underset{x\to 0}{\lim }f\left(x\right)$ doesn’t exist and the complete table is given below:

$x$	$0.1$	$0.01$	$0.001$	$0.0001$
$f\left(x\right)$	$-0.0743$	$0.0099$	$-0.0005$	$-0.00002$

Explanation of Solution

Given information:

The function is $f\left(x\right)=x\sin \left(\ln \left|x\right|\right)$ .

Calculation:

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

  $\begin{array}{c}f\left(x\right)=x\sin \left(\ln \left|x\right|\right)\\ f\left(0.1\right)=0.1\times \sin \left(\ln \left|0.1\right|\right)\\ f\left(0.1\right)=0.1\times \sin \left(\ln \left(0.1\right)\right)\end{array}$

To find the value of $0.1\times \sin \left(\ln \left(0.1\right)\right)$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{0.1}\boxed{\times }\boxed{\text{SIN}}\boxed{\text{LN}}\boxed{0.1}\boxed{)}\boxed{)}$

The value of $0.1\times \sin \left(\ln \left(0.1\right)\right)$ using a calculator is approximately $-0.0743$ .

So, the value of $f\left(x\right)$ at $x=0.1$ is $-0.0743$ .

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

  $\begin{array}{c}f\left(x\right)=x\sin \left(\ln \left|x\right|\right)\\ f\left(0.01\right)=0.01\times \sin \left(\ln \left|0.01\right|\right)\\ f\left(0.01\right)=0.01\times \sin \left(\ln \left(0.01\right)\right)\end{array}$

To find the value of $0.01\times \sin \left(\ln \left(0.01\right)\right)$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{0.01}\boxed{\times }\boxed{\text{SIN}}\boxed{\text{LN}}\boxed{0.01}\boxed{)}\boxed{)}$

The value of $0.01\times \sin \left(\ln \left(0.01\right)\right)$ using a calculator is approximately $0.0099$ .

So, the value of $f\left(x\right)$ at $x=0.01$ is $0.0099$ .

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

  $\begin{array}{c}f\left(x\right)=x\sin \left(\ln \left|x\right|\right)\\ f\left(0.001\right)=0.001\times \sin \left(\ln \left|0.001\right|\right)\\ f\left(0.001\right)=0.001\times \sin \left(\ln \left(0.001\right)\right)\end{array}$

To find the value of $0.001\times \sin \left(\ln \left(0.001\right)\right)$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{0.001}\boxed{\times }\boxed{\text{SIN}}\boxed{\text{LN}}\boxed{0.001}\boxed{)}\boxed{)}$

The value of $0.001\times \sin \left(\ln \left(0.001\right)\right)$ using a calculator is approximately $-0.0005$ .

So, the value of $f\left(x\right)$ at $x=0.001$ is $-0.0005$ .

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

  $\begin{array}{c}f\left(x\right)=x\sin \left(\ln \left|x\right|\right)\\ f\left(0.0001\right)=0.0001\times \sin \left(\ln \left|0.0001\right|\right)\\ f\left(0.0001\right)=0.0001\times \sin \left(\ln \left(0.0001\right)\right)\end{array}$

To find the value of $0.0001\times \sin \left(\ln \left(0.0001\right)\right)$ with the help of calculator press “ON” button and enter the keystrokes as given below:

  $\boxed{0.0001}\boxed{\times }\boxed{\text{SIN}}\boxed{\text{LN}}\boxed{0.0001}\boxed{)}\boxed{)}$

The value of $0.0001\times \sin \left(\ln \left(0.0001\right)\right)$ using a calculator is approximately $-0.00002$ .

So, the value of $f\left(x\right)$ at $x=-0.0001$ is $-0.00002$ .

Therefore, the complete table is given by:

$x$	$0.1$	$0.01$	$0.001$	$0.0001$
$f\left(x\right)$	$-0.0743$	$0.0099$	$-0.0005$	$-0.00002$

The value of the function is not possible to find when $x$ tends to $0$ as the values of function are oscillates with decrease in $x$ .

Therefore, The value of the $\underset{x\to 0}{\lim }f\left(x\right)$ doesn’t exist.