Chapter 11, Problem 36CR

(a)

To determine

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

Answer to Problem 36CR

  $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}=3$

Explanation of Solution

Given information: $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}$ .

Graph:

  

Interpretation:

From the graph, as x gets closer and closer to 0, $\mathrm{f}(\mathrm{x})=\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}$ gets closer and closer to 3 from the left and from the right.

Thus, $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}=3$ .

(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 36CR

  $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}=3$

Explanation of Solution

Given information: $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}$ .

Formula used: Construct a table that shows $\mathrm{f}(\mathrm{x})=\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}$ for two sets of x -values − one set that approaches 0 from the left and one that approaches 0 from the right.

Calculation:

x	$-0.1$	$-0.01$	$-0.001$	0	0.001	0.01	0.1
$\mathrm{f}(\mathrm{x})=\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}$	2.5918177	2.9554466	2.9955	?	3.0045	3.04545	3.498588

From the table,

As x approaches 0 from the left, $\mathrm{f}(\mathrm{x})=\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}$ approaches 3.

As x approaches 0 from the right, $\mathrm{f}(\mathrm{x})=\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}$ approaches 3.

So, it appears that the closer x gets to 0, the closer $\mathrm{f}(\mathrm{x})=\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}$ gets to 3.

Thus, $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{e}}^{3\mathrm{x}}-1}{\mathrm{x}}=3$ .