Chapter 11.2, Problem 65E

a.

To determine

To graph: The piecewise function.

Explanation of Solution

Given: The cost C(in dollars) of making x color photocopies at a copy shop is given by the function

  $\mathrm{C}\left(\mathrm{x}\right)=\{\begin{array}{ccc}0.42\mathrm{x}& & \text{if 0}<\mathrm{x}\le 99\\ 0.32\mathrm{x}& & \text{if 99}<\mathrm{x}\le 999\\ 0.23\mathrm{x}& & \text{if 999}<\mathrm{x}\le 9999\\ 0.16\mathrm{x}& & \text{if }\mathrm{x}>9999\end{array}$

Using graphing calculator to draw the graph of $\mathrm{C}\left(\mathrm{x}\right)$

  

b.

To determine

To find: The limit of each situation.

Explanation of Solution

Given: The cost C(in dollars) of making x color photocopies at a copy shop is given by the function

  $\mathrm{C}\left(\mathrm{x}\right)=\{\begin{array}{ccc}0.42\mathrm{x}& & \text{if 0}<\mathrm{x}\le 99\\ 0.32\mathrm{x}& & \text{if 99}<\mathrm{x}\le 999\\ 0.23\mathrm{x}& & \text{if 999}<\mathrm{x}\le 9999\\ 0.16\mathrm{x}& & \text{if }\mathrm{x}>9999\end{array}$

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

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to 15}{\lim }0.42\mathrm{x}\\ \mathrm{L}=0.42\left(15\right)\\ \mathrm{L}=6.3\end{array}$

  $\mathrm{L}=\underset{\mathrm{x}\to 450}{\lim }\mathrm{C}\left(\mathrm{x}\right)$

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to 450}{\lim }0.32\mathrm{x}\\ \mathrm{L}=0.32\left(450\right)\\ \mathrm{L}=144\end{array}$

  $\mathrm{L}=\underset{\mathrm{x}\to 6000}{\lim }\mathrm{C}\left(\mathrm{x}\right)$

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to 6000}{\lim }0.23\mathrm{x}\\ \mathrm{L}=0.23\left(6000\right)\\ \mathrm{L}=1380\end{array}$

c.

To determine

To make: A table of values to show numerically that limit does not exist.

Answer to Problem 65E

  $\begin{array}{cccc}\text{Function}& \text{LHL}& \text{RHL}& \text{Conclusion}\\ \underset{\mathrm{x}\to 99}{\lim }& 41.58& 31.68& 41.58\ne 31.68\\ \underset{\mathrm{x}\to 999}{\lim }& 319.68& 229.77& 319.68\ne 229.77\\ \underset{\mathrm{x}\to 9999}{\lim }& 2299.77& 1599.84& 2299.77\ne 1599.84\end{array}$

Explanation of Solution

Given: The cost C(in dollars) of making x color photocopies at a copy shop is given by the function

  $\mathrm{C}\left(\mathrm{x}\right)=\{\begin{array}{ccc}0.42\mathrm{x}& & \text{if 0}<\mathrm{x}\le 99\\ 0.32\mathrm{x}& & \text{if 99}<\mathrm{x}\le 999\\ 0.23\mathrm{x}& & \text{if 999}<\mathrm{x}\le 9999\\ 0.16\mathrm{x}& & \text{if }\mathrm{x}>9999\end{array}$

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

Left Hand Limit, LHL

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to {99}^{-}}{\lim }0.42\mathrm{x}\\ \mathrm{L}=0.42\left(99\right)\\ \mathrm{L}=41.58\end{array}$

Right Hand Limit, RHL

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to {99}^{+}}{\lim }0.32\mathrm{x}\\ \mathrm{L}=0.32\left(99\right)\\ \mathrm{L}=31.68\end{array}$

  $\mathrm{L}\mathrm{H}\mathrm{L}\ne \mathrm{R}\mathrm{H}\mathrm{L}$

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

Left Hand Limit, LHL

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to {999}^{-}}{\lim }0.32\mathrm{x}\\ \mathrm{L}=0.32\left(999\right)\\ \mathrm{L}=319.68\end{array}$

Right Hand Limit, RHL

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to {999}^{+}}{\lim }0.23\mathrm{x}\\ \mathrm{L}=0.23\left(999\right)\\ \mathrm{L}=229.77\end{array}$

  $\mathrm{L}\mathrm{H}\mathrm{L}\ne \mathrm{R}\mathrm{H}\mathrm{L}$

  $\mathrm{L}=\underset{\mathrm{x}\to 9999}{\lim }\mathrm{C}\left(\mathrm{x}\right)$

Left Hand Limit, LHL

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to {9999}^{-}}{\lim }0.23\mathrm{x}\\ \mathrm{L}=0.23\left(9999\right)\\ \mathrm{L}=2299.77\end{array}$

Right Hand Limit, RHL

  $\begin{array}{l}\mathrm{L}=\underset{\mathrm{x}\to {9999}^{+}}{\lim }0.16\mathrm{x}\\ \mathrm{L}=0.16\left(9999\right)\\ \mathrm{L}=1599.84\end{array}$

  $\mathrm{L}\mathrm{H}\mathrm{L}\ne \mathrm{R}\mathrm{H}\mathrm{L}$

  $\begin{array}{cccc}\text{Function}& \text{LHL}& \text{RHL}& \text{Conclusion}\\ \underset{\mathrm{x}\to 99}{\lim }& 41.58& 31.68& 41.58\ne 31.68\\ \underset{\mathrm{x}\to 999}{\lim }& 319.68& 229.77& 319.68\ne 229.77\\ \underset{\mathrm{x}\to 9999}{\lim }& 2299.77& 1599.84& 2299.77\ne 1599.84\end{array}$

d.

To determine

To verufy: The limit value in part (c) from the graph in part (a).

Explanation of Solution

Given: The cost C(in dollars) of making x color photocopies at a copy shop is given by the function

  $\mathrm{C}\left(\mathrm{x}\right)=\{\begin{array}{ccc}0.42\mathrm{x}& & \text{if 0}<\mathrm{x}\le 99\\ 0.32\mathrm{x}& & \text{if 99}<\mathrm{x}\le 999\\ 0.23\mathrm{x}& & \text{if 999}<\mathrm{x}\le 9999\\ 0.16\mathrm{x}& & \text{if }\mathrm{x}>9999\end{array}$

Graph from part (a)

  

Table from part (c)

  $\begin{array}{cccc}\text{Function}& \text{LHL}& \text{RHL}& \text{Conclusion}\\ \underset{\mathrm{x}\to 99}{\lim }& 41.58& 31.68& 41.58\ne 31.68\\ \underset{\mathrm{x}\to 999}{\lim }& 319.68& 229.77& 319.68\ne 229.77\\ \underset{\mathrm{x}\to 9999}{\lim }& 2299.77& 1599.84& 2299.77\ne 1599.84\end{array}$

At each break points graph break. It seen limit does not exixt at these points.