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Estimate the limit numerically or state that the limit does not exist.

[tex]\[ \lim_{x \rightarrow 0} \frac{7^x - 1}{x} \][/tex]

Start by calculating the values of the function for the inputs listed in this table.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
0.01 & \\
\hline
0.001 & \\
\hline
0.0001 & \\
\hline
-0.0001 & \\
\hline
-0.001 & \\
\hline
-0.01 & \\
\hline
\end{tabular}

Based on the values in this table, it appears

[tex]\[ \lim_{x \rightarrow 0} \frac{7^x - 1}{x} = \square \][/tex]

Give your answer accurate to 3 decimal places.


Sagot :

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]