Chapter 11.2, Problem 41E

To determine

To calculate: The limit, $\underset{x\to 0}{\lim }{\left(1+x\right)}^{\frac{3}{x}}$ upto three decimal places using the TABLE feature of the graphing utility.

Answer to Problem 41E

The limit, $\underset{x\to 0}{\lim }{\left(1+x\right)}^{\frac{3}{x}}$ is 20.090.

Explanation of Solution

Given: The function and point are ${\left(1+x\right)}^{\frac{3}{x}}$ and $x=0$ respectively.

Calculation: Click on $\text{Y}=$ key to enter the given function.

  ${\text{Y}}_1=\left(1+\text{X}\right)^\left(3/\text{X}\right)$

Click on 2nd key followed by WINDOW key and change the default setting of “Indpnt” variable to “Ask”.

Click on 2nd key followed by GRAPH key and enter the values of x for which the value of function is needed.

x	-0.1	-0.01	-0.001	0	0.001	0.01	0.1
${\text{Y}}_1$	23.59	20.391	20.116	ERROR	20.055	19.788	17.449

As the value of the function near 0 is approximately equal, the limit exists.

Find the average of 20.116 and 20.055 to get the limit of the function when x approaches to 0.

  $\begin{array}{c}\frac{20.166+20.055}{2}=\frac{40.171}{2}\\ \approx 20.090\end{array}$

Thus, the value of $\underset{x\to 0}{\lim }{\left(1+x\right)}^{\frac{3}{x}}$ is 20.090.