Chapter 11.2, Problem 65E

a.

To determine

To graph: the following function,

  $C\left(x\right)=\{\begin{array}{l}0.20x,0<x\le 9\\ 0.15x,9<x\le 99\\ 0.10x,99<x\le 999\\ 0.08x,x>999\end{array}$

Explanation of Solution

Given information:

The cost C of making x photocopies at a copy shop is given by the function,

  $C\left(x\right)=\{\begin{array}{l}0.20x,0<x\le 9\\ 0.15x,9<x\le 99\\ 0.10x,99<x\le 999\\ 0.08x,x>999\end{array}$ .

Graph:

The graph of function $C\left(x\right)$ is,

  

b.

To determine

To identify: The limit of the following:

  $\underset{x\to 15}{\lim }C\left(x\right)$

  $\underset{x\to 98}{\lim }C\left(x\right)$

  $\underset{x\to 405}{\lim }C\left(x\right)$

Answer to Problem 65E

  $\begin{array}{l}\underset{x\to 15}{\lim }C\left(x\right)=2.25\\ \underset{x\to 98}{\lim }C\left(x\right)=14.7\\ \underset{x\to 405}{\lim }C\left(x\right)=40.5\end{array}$

Explanation of Solution

Given information:

The function,

  $C\left(x\right)=\{\begin{array}{l}0.20x,0<x\le 9\\ 0.15x,9<x\le 99\\ 0.10x,99<x\le 999\\ 0.08x,x>999\end{array}$

  $\begin{array}{c}\underset{x\to 15}{\lim }C\left(x\right)=\underset{x\to 15}{\lim }\left(0.15x\right)\\ =0.15\times 15\\ =2.25\end{array}$

  $\begin{array}{c}\underset{x\to 98}{\lim }C\left(x\right)=\underset{x\to 98}{\lim }\left(0.15x\right)\\ =0.15\times 98\\ =14.7\end{array}$

  $\begin{array}{c}\underset{x\to 405}{\lim }C\left(x\right)=\underset{x\to 405}{\lim }\left(0.10x\right)\\ =0.10\times 405\\ =40.5\end{array}$

Conclusion:

  $\begin{array}{l}\underset{x\to 15}{\lim }C\left(x\right)=2.25\\ \underset{x\to 98}{\lim }C\left(x\right)=14.7\\ \underset{x\to 405}{\lim }C\left(x\right)=40.5\end{array}$

c.

To determine

To identify: The limit of the following function does not exist:

  $\underset{x\to 9}{\lim }C\left(x\right)$

  $\underset{x\to 99}{\lim }C\left(x\right)$

  $\underset{x\to 999}{\lim }C\left(x\right)$

Answer to Problem 65E

The limit of cost function $C\left(x\right)$ does not exist at $x=9,99,999$ .

Explanation of Solution

Given information:

The function,

  $C\left(x\right)=\{\begin{array}{l}0.20x,0<x\le 9\\ 0.15x,9<x\le 99\\ 0.10x,99<x\le 999\\ 0.08x,x>999\end{array}$

Concept Involved:

The limit of a function $f\left(x\right)$ at x = a does not exist if,

Either

  $\underset{x\to a^{-}}{\lim }f\left(x\right)\ne \underset{x\to a^{+}}{\lim }f\left(x\right)$

Or

  $\underset{x\to a}{\lim }f\left(x\right)\ne \underset{x\to a^{-}}{\lim }f\left(x\right)\text{or}\underset{x\to a^{+}}{\lim }f\left(x\right)$

Create a table to find the limit of $C\left(x\right)$ .

  $\begin{array}{ccc}\text{Right}\text{hand}\text{limit}& \text{Left}\text{hand}\text{limit}& \text{Function}\text{limit}\\ \underset{x\to 9^{+}}{\lim }C\left(x\right)=1.35& \underset{x\to 9^{-}}{\lim }C\left(x\right)=1.80& \underset{x\to 9}{\lim }C\left(x\right)=1.80\\ \underset{x\to {99}^{+}}{\lim }C\left(x\right)=9.9& \underset{x\to {99}^{-}}{\lim }C\left(x\right)=14.85& \underset{x\to 99}{\lim }C\left(x\right)=14.85\\ \underset{x\to {999}^{+}}{\lim }C\left(x\right)=79.92& \underset{x\to {999}^{-}}{\lim }C\left(x\right)=99.9& \underset{x\to 999}{\lim }C\left(x\right)=99.9\end{array}$

From the table shows that

  $\begin{array}{l}\underset{x\to 9^{+}}{\lim }C\left(x\right)\ne \underset{x\to 9^{-}}{\lim }C\left(x\right)=\underset{x\to 9}{\lim }C\left(x\right)\\ \underset{x\to {99}^{+}}{\lim }C\left(x\right)\ne \underset{x\to {99}^{-}}{\lim }C\left(x\right)=\underset{x\to 99}{\lim }C\left(x\right)\\ \underset{x\to {999}^{+}}{\lim }C\left(x\right)\ne \underset{x\to {999}^{-}}{\lim }C\left(x\right)=\underset{x\to 999}{\lim }C\left(x\right)\end{array}$

This implies that the limit of $C\left(x\right)$ does not exist at $x=9,99,999$ .

Conclusion:

The limit of cost function $C\left(x\right)$ does not exist at $x=9,99,999$ .

d.

To determine

To identify: from the graph that the limits of following function do not exist.

1. $\underset{x\to 9}{\lim }C\left(x\right)$
2. $\underset{x\to 99}{\lim }C\left(x\right)$
3. $\underset{x\to 999}{\lim }C\left(x\right)$

Explanation of Solution

Given information:

The function,

  $C\left(x\right)=\{\begin{array}{l}0.20x,0<x\le 9\\ 0.15x,9<x\le 99\\ 0.10x,99<x\le 999\\ 0.08x,x>999\end{array}$

In order to find the limit of a function is exist or not, the function must approach the same value from the left and from the right.

From the graph in part (a),

At $x=9$ , the cost function $C\left(x\right)$ gives the different values from the right and left points.

So, the limit function $\underset{x\to 9}{\lim }C\left(x\right)$ does not exist.

Similarly,

At $x=99$ , the cost function $C\left(x\right)$ gives the different values from the right and left points.

And

At $x=999$ , the cost function $C\left(x\right)$ gives the different values from the right and left points.

So, the limit functions $\underset{x\to 99}{\lim }C\left(x\right)$ and $\underset{x\to 999}{\lim }C\left(x\right)$ do not exist.

Conclusion:

The limit of cost function $C\left(x\right)$ does not exist at $x=9,99,999$ .