Chapter 11, Problem 33CR

(a)

To determine

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

Answer to Problem 33CR

  $\underset{\mathrm{x}\to 3}{\lim }\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}=\frac{1}{6}$

Explanation of Solution

Given information: $\underset{\mathrm{x}\to 3}{\lim }\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}$ .

Graph:

  

Interpretation:

From the graph, as x gets closer and closer to 3, $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}$ gets closer and closer to $\frac{1}{6}$ from the left and from the right.

Thus, $\underset{\mathrm{x}\to 3}{\lim }\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}=\frac{1}{6}$ .

(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 33CR

  $\underset{\mathrm{x}\to 3}{\lim }\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}=\frac{1}{6}$

Explanation of Solution

Given information: $\underset{\mathrm{x}\to 3}{\lim }\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}$ .

Formula used: Construct a table that shows $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}$ for two sets of x -values − one set that approaches 3 from the left and one that approaches 3 from the right.

Calculation:

x	2.9	2.99	2.999	3	3.001	3.01	3.1
$\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}$	0.169	0.1669	0.16669	?	0.16663	0.1663	0.163

From the table,

As x approaches 3 from the left, $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}$ approaches $\frac{1}{6}$ .

As x approaches 3 from the right, $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}$ approaches $\frac{1}{6}$ .

So, it appears that the closer x gets to 3, the closer $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}$ gets to $\frac{1}{6}$ .

Thus, $\underset{\mathrm{x}\to 3}{\lim }\frac{\mathrm{x}-3}{{\mathrm{x}}^2-9}=\frac{1}{6}$ .