Chapter 11, Problem 34CR

(a)

To determine

To find: The limit (if it exists) graphically by using graphing utility to graph the function.

Answer to Problem 34CR

  $\underset{\mathrm{x}\to 4}{\lim }\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}=\frac{1}{8}$

Explanation of Solution

Given information: $\underset{\mathrm{x}\to 4}{\lim }\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}$ .

Graph:

  

Interpretation:

From the graph, as x gets closer and closer to 4, $\mathrm{f}(\mathrm{x})=\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}$ gets closer and closer to $\frac{1}{8}$ from the left and from the right.

Thus, $\underset{\mathrm{x}\to 4}{\lim }\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}=\frac{1}{8}$ .

(b)

To determine

To find: The limit (if it exists) numerically by using the table feature of the graphing utility to create a table.

Answer to Problem 34CR

  $\underset{\mathrm{x}\to 4}{\lim }\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}=\frac{1}{8}$

Explanation of Solution

Given information: $\underset{\mathrm{x}\to 4}{\lim }\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}$ .

Formula used: Construct a table that shows $\mathrm{f}(\mathrm{x})=\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}$ for two sets of x -values − one set that approaches 4 from the left and one that approaches 4 from the right.

Calculation:

x	3.9	3.99	3.999	4	4.001	4.01	4.1
$\mathrm{f}(\mathrm{x})=\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}$	0.12658	0.125156	0.1250156	?	0.124984	0.1248439	0.12345679

From the table,

As x approaches 4 from the left, $\mathrm{f}(\mathrm{x})=\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}$ approaches $\frac{1}{8}$ .

As x approaches 4 from the right, $\mathrm{f}(\mathrm{x})=\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}$ approaches $\frac{1}{8}$ .

So, it appears that the closer x gets to 4, the closer $\mathrm{f}(\mathrm{x})=\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}$ gets to $\frac{1}{8}$ .

Thus, $\underset{\mathrm{x}\to 4}{\lim }\frac{4-\mathrm{x}}{16-{\mathrm{x}}^2}=\frac{1}{8}$ .