Chapter 2.1, Problem 78E

(a)

To determine

To show: The value of $\underset{x\to 2}{\lim }f\left(x\right)=2=f\left(2\right)$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=\sqrt{3x-2}$ .

Proof:

As given the function is $f\left(x\right)=\sqrt{3x-2}$ , this function corresponds for both the left and right limits of $f\left(x\right)$ .

So, $\underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }f\left(x\right)=\underset{x\to 2}{\lim }f\left(x\right)$ .

Simplify the limit.

  $\begin{array}{c}\underset{x\to 2}{\lim }f\left(x\right)=\underset{x\to 2}{\lim }\left(\sqrt{3x-2}\right)\\ =\sqrt{3\times 2-2}\\ =\sqrt{6-2}\\ =2\end{array}$

So, the value of $\underset{x\to 2}{\lim }\left(\sqrt{3x-2}\right)$ is equal to $2$ .

Now, substitute $2$ for $x$ in the function.

  $\begin{array}{l}f\left(x\right)=\sqrt{3x-2}\\ f\left(2\right)=\sqrt{3\times 2-2}\\ f\left(2\right)=\sqrt{6-2}\\ f\left(2\right)=2\end{array}$

Hence, it is proved that $\underset{x\to 2}{\lim }f\left(x\right)=2=f\left(2\right)$ .

(b)

To determine

To find: The interval $\left(a,b\right)$ so that $1.8<f\left(x\right)<2.2$ with the help of graph provided $a<x<b$ .

Answer to Problem 78E

The value of $a$ and $b$ are $1.75$ and $2.28$ respectively.

Explanation of Solution

Given information: The function is $f\left(x\right)=\sqrt{3x-2}$ .

Calculation:

The point $a$ is the intersection of the curve $f\left(x\right)=\sqrt{3x-2}$ and $y=1.8$ , the point $b$ is the intersection of the curve $f\left(x\right)=\sqrt{3x-2}$ and $y=2.2$ .

Graph the curves.

To graph a function $f\left(x\right)=\sqrt{3x-2}$ , $y=1.8$ and $y=2.2$ , 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 $f\left(x\right)=\sqrt{3x-2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{3}\boxed{\text{X}}\boxed{-}\boxed{2}\boxed{)}\boxed{^}\boxed{1}\boxed{÷}\boxed{2}$ . Now press Enter, enter right hand side of the equation $y=1.8$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{1.8}$ . Now press Enter, enter right hand side of the equation $y=2.2$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes $\boxed{2.2}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\left(3\text{X}-2\right)^1/2\\ {\text{Y}}_2=1.8\\ {\text{Y}}_3=2.2\end{array}$

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)

The curves $f\left(x\right)=\sqrt{3x-2}$ and $y=1.8$ intersects at the point $x=1.7467$ and the curves $f\left(x\right)=\sqrt{3x-2}$ and $y=2.2$ intersects at the point $x=2.28$ .

Therefore, the value of $a$ and $b$ are $1.75$ and $2.28$ respectively.

(c)

To determine

To find: The interval $\left(a,b\right)$ so that $1.99<f\left(x\right)<2.01$ with the help of graph provided $a<x<b$ .

Answer to Problem 78E

The value of $a$ and $b$ are $1.9867$ and $2.0134$ respectively.

Explanation of Solution

Given information: The function is $f\left(x\right)=\sqrt{3x-2}$ .

Calculation:

The point $a$ is the intersection of the curve $f\left(x\right)=\sqrt{3x-2}$ and $y=1.99$ , the point $b$ is the intersection of the curve $f\left(x\right)=\sqrt{3x-2}$ and $y=2.01$ .

Graph the curves.

To graph a function $f\left(x\right)=\sqrt{3x-2}$ , $y=1.99$ and $y=2.01$ , 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 $f\left(x\right)=\sqrt{3x-2}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{3}\boxed{\text{X}}\boxed{-}\boxed{2}\boxed{)}\boxed{^}\boxed{1}\boxed{÷}\boxed{2}$ . Now press Enter, enter right hand side of the equation $y=1.99$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{1.99}$ . Now press Enter, enter right hand side of the equation $y=2.01$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes $\boxed{2.01}$ .

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\left(3\text{X}-2\right)^1/2\\ {\text{Y}}_2=1.99\\ {\text{Y}}_3=2.01\end{array}$

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)

The curves $f\left(x\right)=\sqrt{3x-2}$ and $y=1.99$ intersects at the point $x=1.9867$ and the curves $f\left(x\right)=\sqrt{3x-2}$ and $y=2.01$ intersects at the point $x=2.0134$ .

Therefore, the value of $a$ and $b$ are $1.9867$ and $2.0134$ respectively.