Chapter 2.1, Problem 9E

To determine

To find: The value of $\underset{x\to 1}{\lim }\left(x^3+3x^2-2x-17\right)$ with the help of substitution. Check the results graphically.

Answer to Problem 9E

The value of $\underset{x\to 1}{\lim }\left(x^3+3x^2-2x-17\right)$ is $-15$ .

Explanation of Solution

Given information:

The given limit is $\underset{x\to 1}{\lim }\left(x^3+3x^2-2x-17\right)$ .

Calculation:

Substitute $1$ for $x$ in the given limit.

  $\begin{array}{c}\underset{x\to 1}{\lim }\left(x^3+3x^2-2x-17\right)=1^3+3\times 1^2-2\times 1-17\\ =1+3-2-17\\ =-15\end{array}$

Therefore, the value of $\underset{x\to 1}{\lim }\left(x^3+3x^2-2x-17\right)$ is $-15$ .

Check the limit graphically:

Follow the steps using graphing calculator to graph a function $y=x^3+3x^2-2x-17$ .

First press “ON” button on graphical calculator. Press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=x^3+3x^2-2x-17$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{^}\boxed{3}\boxed{+}\boxed{3}\boxed{\text{X}}\boxed{^}\boxed{2}\boxed{-}\boxed{2}\boxed{\text{X}}\boxed{-}\boxed{17}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\text{X}+3\right)^1998$

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 $\boxed{2\text{nd}}\boxed{\text{TRACE}}\boxed{1}\boxed{1}$ 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 $\underset{x\to 1}{\lim }\left(x^3+3x^2-2x-17\right)$ is $-15$ .