Chapter 2.1, Problem 13E

To determine

To find: The value of $\underset{x\to -2}{\lim }{\left(x-6\right)}^{\frac{2}{3}}$ with the help of substitution. Check the results graphically.

Answer to Problem 13E

The value of $\underset{x\to -2}{\lim }{\left(x-6\right)}^{\frac{2}{3}}$ is $4$ .

Explanation of Solution

Given information:

The given limit is $\underset{x\to -2}{\lim }{\left(x-6\right)}^{\frac{2}{3}}$ .

Calculation:

Substitute $-2$ for $x$ in the given limit.

  $\begin{array}{c}\underset{x\to -2}{\lim }{\left(x-6\right)}^{\frac{2}{3}}={\left(-2-6\right)}^{\frac{2}{3}}\\ ={\left({\left(-8\right)}^2\right)}^{\frac{1}{3}}\\ ={\left(64\right)}^{\frac{1}{3}}\\ =4\end{array}$

Therefore, the value of $\underset{x\to -2}{\lim }{\left(x-6\right)}^{\frac{2}{3}}$ is $4$ .

Check the limit graphically:

Follow the steps using graphing calculator to graph a function $y={\left(x-6\right)}^{\frac{2}{3}}$ .

First press “ON” button on graphical calculator. Press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y={\left(x-6\right)}^{\frac{2}{3}}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{6}\boxed{)}\boxed{^}\boxed{(}\boxed{2}\boxed{÷}\boxed{3}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\text{X}-6\right)^\left(2/3\right)$

Press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function. Enter the keystrokes for the value of the function at particular point $\boxed{2\text{nd}}\boxed{\text{TRACE}}\boxed{1}\boxed{(-)}\boxed{2}$ and press the $\boxed{\text{ENTER}}$ key.

The graph plotted in standard window as shown in Figure (1).

  

Figure (1)

It is verified graphically that the value of $\underset{x\to -2}{\lim }{\left(x-6\right)}^{\frac{2}{3}}$ is $4$ .