To solve the limit \( \lim_{\theta \to 0} \frac{\tan(3\theta)}{4\theta} \), we can approach this using the known limit \( \lim_{\theta \to 0} \frac{\tan(\theta)}{\theta} = 1 \).

### Steps:

1. **Substitute the argument of the tangent:**

\[
\lim_{\theta \to 0} \frac{\tan(3\theta)}{4\theta}
\]

2. **Use the limit identity for small angles:**

We know that:
\[
\lim_{\theta \to 0} \frac{\tan(\theta)}{\theta} = 1
\]
Let's rewrite the limit in a form that uses this identity:
\[
\lim_{\theta \to 0} \frac{\tan(3\theta)}{3\theta} \times \frac{3\theta}{4\theta}
\]

This simplifies to:
\[
\lim_{\theta \to 0} \left( \frac{\tan(3\theta)}{3\theta} \times \frac{3}{4} \right)
\]

3. **Apply the limit identity:**

\[
\lim_{\theta \to 0} \frac{\tan(3\theta)}{3\theta} = 1
\]

Thus, the limit becomes:
\[
1 \times \frac{3}{4} = \frac{3}{4}
\]

### Conclusion:
\[
\lim_{\theta \to 0} \frac{\tan(3\theta)}{4\theta} = \frac{3}{4}
\]

Would you like to explore this solution further?

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Here are 8 related questions:

1. How does the small-angle approximation help in calculating limits involving trigonometric functions?
2. What is the geometric interpretation of \( \lim_{\theta \to 0} \frac{\tan(\theta)}{\theta} = 1 \)?
3. How would you compute limits of trigonometric functions without using known identities?
4. What happens if the angle inside the tangent function is different, like \( 5\theta \)?
5. Can you explain L'Hopital's rule and how it applies to trigonometric limits?
6. What is the significance of limits in calculus?
7. How would this limit change if the denominator were \( 5\theta \) instead of \( 4\theta \)?
8. How would you calculate \( \lim_{\theta \to 0} \frac{\sin(3\theta)}{4\theta} \)?

**Tip:** When dealing with trigonometric limits, applying known limits like \( \frac{\sin(\theta)}{\theta} \) and \( \frac{\tan(\theta)}{\theta} \) for small angles can simplify the problem significantly.

Explore how to solve the limit \( \lim_{\theta \to 0} \frac{\tan(3\theta)}{4\theta} \) using trigonometric identities and limits. This problem involves the application of the small-angle approximation and known limits of trigonometric functions. Suitable for advanced high school students and those studying calculus.

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