The expression \(\frac{\sin(3x)}{x}\) is undefined at \(x = 0\) because division by zero is not allowed. However, we can explore the behavior of this function for small values of \(x\), particularly as \(x\) approaches 0, which can be useful in calculus.

### Finding the Limit as \(x\) Approaches 0:

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

To evaluate this limit, we can use the standard limit result:

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

In this case, \(k = 3\), so:

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

Thus, the limit of \(\frac{\sin(3x)}{x}\) as \(x\) approaches 0 is 3.

Would you like to explore further details, or do you have any questions?

Here are 5 related questions:
1. How would you compute \(\frac{\sin(kx)}{x}\) for different values of \(k\)?
2. What is the derivative of \(\frac{\sin(3x)}{x}\)?
3. How does the graph of \(\frac{\sin(3x)}{x}\) behave as \(x\) increases?
4. Can you expand \(\sin(3x)\) using a Taylor series around \(x = 0\)?
5. What happens to \(\frac{\sin(3x)}{x}\) for large values of \(x\)?

**Tip**: Knowing common limits like \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\) can help in evaluating more complex limits efficiently.

Explore the step-by-step solution to find the limit of (sin(3x))/x as x approaches 0. This problem involves key trigonometric concepts, limits, and calculus techniques. Learn how to use the Sine Limit Theorem and related formulas to solve this problem, suitable for Grades 11-12 students.

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