To evaluate the limit [tex]\(\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}\)[/tex] using given values of [tex]\(x\)[/tex], we can set up a table with the corresponding values of [tex]\(\frac{\sin 3 x}{x}\)[/tex].
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & \frac{\sin 3 x}{x} & x & \frac{\sin 3 x}{x} \\
\hline
-0.1 & 2.95520207 & 0.1 & 2.95520207 \\
\hline
-0.01 & 2.99955002 & 0.01 & 2.99955002 \\
\hline
-0.001 & 2.99999550 & 0.001 & 2.99999550 \\
\hline
-0.0001 & 2.99999996 & 0.0001 & 2.99999996 \\
\hline
\end{array}
\][/tex]
Based on the table, as [tex]\(x\)[/tex] approaches 0 from both the negative and positive sides, the value of [tex]\(\frac{\sin 3 x}{x}\)[/tex] gets closer to approximately 3. Consequently, we can estimate that
[tex]\[
\lim _{x \rightarrow 0} \frac{\sin 3 x}{x} \approx 3.
\][/tex]
