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The given expression asks for the limit:

\[
\lim_{{x \to 0}} \frac{\sin(3x)}{x}
\]

To solve this limit, we can use a standard trigonometric limit result:

\[
\lim_{{x \to 0}} \frac{\sin(kx)}{x} = k
\]

for any constant \( k \). In our case, \( k = 3 \), so:

\[
\lim_{{x \to 0}} \frac{\sin(3x)}{x} = 3
\]

Thus, the value of the limit is:

\[
\boxed{3}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the limit \(\lim_{{x \to 0}} \frac{\sin(x)}{x} = 1\) help in this problem?
2. What is the geometric interpretation of the limit of \(\frac{\sin(x)}{x}\) as \(x\) approaches 0?
3. Can L'Hôpital's Rule be used to solve this limit, and if so, how?
4. How do limits involving trigonometric functions behave near 0?
5. What other common limits should be memorized for solving calculus problems?

**Tip:** Memorizing key trigonometric limits, such as \(\lim_{{x \to 0}} \frac{\sin(x)}{x} = 1\), can save time in solving many limit problems!

Evaluate the limit: lim (x -> 0) sin(3x) / x

Learn how to evaluate the limit lim (x -> 0) sin(3x) / x. This problem covers the concept of limits and the standard trigonometric limit result lim (x -> 0) sin(kx) / x = k, where k is a constant. Suitable for high school students in Grades 10-12.

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