Chapter 11.1, Problem 24E

To determine

To create: a table for the given function and find the limit and graph the function.

Answer to Problem 24E

x	$f\left(x\right)$
-0.1	4.91825
-0.01	4.08108
-0.001	4.00801
0	Indeterminate
0.001	3.99201
0.01	3.92106
0.1	3.2968

  $\underset{x\to 0}{\lim }\frac{1-e^{-4x}}{x}=4$

Explanation of Solution

Given:

  $\underset{x\to 0}{\lim }\frac{1-e^{-4x}}{x}$

Calculation:

Let $f\left(x\right)=\frac{1-e^{-4x}}{x}$

Use graphing calculator to create a table of values of $f\left(x\right)$ for x around 0, and see if the function appears to approach a particular value as x approaches 0 from both the left and right.

x	$f\left(x\right)$
-0.1	4.91825
-0.01	4.08108
-0.001	4.00801
0	Indeterminate
0.001	3.99201
0.01	3.92106
0.1	3.2968

Here, $f\left(x\right)$ takes values close to 4 as x takes values close to 0, even though $f\left(x\right)$ is not defined there. Thus, $\underset{x\to 0}{\lim }f\left(x\right)=4$ . This can also be seen from the graph:

  

Conclusion:

Therefore, the value of the given function is $\underset{x\to 0}{\lim }\frac{1-e^{-4x}}{x}=4$

The table is created for the given function.