To solve the limit [tex]\(\lim_{x \to 0} \frac{7^x - 1}{x}\)[/tex] numerically, we will calculate the values of the function for inputs close to 0, both positive and negative. The function to evaluate is:
[tex]\[f(x) = \frac{7^x - 1}{x}\][/tex]
The selected inputs for [tex]\(x\)[/tex] are: 0.01, 0.001, 0.0001, -0.0001, -0.001, and -0.01. We will fill in the following table with these values:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
0.01 & 1.9649663856459165 \\
\hline
0.001 & 1.9478046608600863 \\
\hline
0.0001 & 1.9460994896514094 \\
\hline
-0.0001 & 1.945720833019804 \\
\hline
-0.001 & 1.9440180933569806 \\
\hline
-0.01 & 1.927099527708498 \\
\hline
\end{tabular}
\][/tex]
Now, looking at the values in the table, we see that as [tex]\( x \)[/tex] gets closer to 0, both from the positive and negative sides, the values of [tex]\( f(x) \)[/tex] appear to be approaching a certain number.
To confirm the limit more precisely, we observe the value of [tex]\( f(x) \)[/tex] for [tex]\( x = 0.0001 \)[/tex] which is very close to 0. The value [tex]\( f(0.0001) \)[/tex] is approximately equal to 1.9460994896514094. This is the closest point we have calculated near zero, and it provides a good estimate for the limit.
Based on the computed values, we can conclude that the limit is:
[tex]\[
\lim_{x \to 0} \frac{7^x - 1}{x} \approx 1.946
\][/tex]
Thus, the limit accurate to three decimal places is:
[tex]\[
\boxed{1.946}
\][/tex]
