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(10 points) Use the given values of [tex]$x$[/tex] to set up a table to evaluate the limit. Round your solutions to eight decimal places.

[tex]
\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}
[/tex]

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$\frac{\sin 3 x}{x}$[/tex] & [tex]$x$[/tex] & [tex]$\frac{\sin 3 x}{x}$[/tex] \\
\hline
-0.1 & & 0.1 & \\
\hline
-0.01 & & 0.01 & \\
\hline
-0.001 & & 0.001 & \\
\hline
-0.0001 & & 0.0001 & \\
\hline
\end{tabular}

Give your estimate of the value of the limit.


Sagot :

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]