To complete the table and estimate the limit, we will evaluate the function \( f(x) = \frac{3 \sin(x)}{x} \) at the specified values of \( x \) and round the answers to five decimal places.
Given the values of \( x \):
- \( x = -0.1 \)
- \( x = -0.01 \)
- \( x = -0.001 \)
- \( x = 0 \) (noting that direct substitution leads to an indeterminate form, but using limit properties)
- \( x = 0.001 \)
- \( x = 0.01 \)
- \( x = 0.1 \)
We can plug these into the function to get:
[tex]\[
f(-0.1) \approx 2.99500
\][/tex]
[tex]\[
f(-0.01) \approx 2.99995
\][/tex]
[tex]\[
f(-0.001) \approx 3.00000
\][/tex]
[tex]\[
f(0) = 3 \quad \text{(limit as } x \text{ approaches 0)}
\][/tex]
[tex]\[
f(0.001) \approx 3.00000
\][/tex]
[tex]\[
f(0.01) \approx 2.99995
\][/tex]
[tex]\[
f(0.1) \approx 2.99500
\][/tex]
Arranging these values in the table, we get:
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -0.1 & -0.01 & -0.001 & 0 & 0.001 & 0.01 & 0.1 \\\hline
[tex]$f(x)$[/tex] & 2.99500 & 2.99995 & 3.00000 & 3 & 3.00000 & 2.99995 & 2.99500 \\
\hline
\end{tabular}
To estimate the limit of \( \frac{3 \sin(x)}{x} \) as \( x \) approaches 0, we observe the values in the table. They are approaching 3 from both sides as \( x \) approaches 0.
Therefore, we estimate:
[tex]\[
\lim_{x \rightarrow 0} \frac{3 \sin(x)}{x} \approx 3.00000
\][/tex]
Graphing the function [tex]\( f(x) = \frac{3 \sin(x)}{x} \)[/tex] would confirm that it approaches 3 as [tex]\( x \)[/tex] approaches 0, which aligns with our numerical estimation.
