We are tasked with evaluating the limit:

$\lim_{\theta \to 0} \frac{\sin(3\theta) \cdot \tan(4\theta)}{\theta^2}$

Step-by-step solution:

1. Approximate $\sin(3\theta)$ and $\tan(4\theta)$ for small $\theta$:

   As $\theta \to 0$, both $\sin(x)$ and $\tan(x)$ behave approximately as $x$. Thus:

   • $\sin(3\theta) \approx 3\theta$
   • $\tan(4\theta) \approx 4\theta$

2. Substitute these approximations into the limit:

   Substituting $\sin(3\theta) \approx 3\theta$ and $\tan(4\theta) \approx 4\theta$, the expression becomes:

   $\lim_{\theta \to 0} \frac{(3\theta) \cdot (4\theta)}{\theta^2}$

3. Simplify the expression:

   Simplifying the numerator:

   $3\theta \cdot 4\theta = 12\theta^2$

   Therefore, the expression becomes:

   $\lim_{\theta \to 0} \frac{12\theta^2}{\theta^2}$

4. Cancel $\theta^2$ from the numerator and denominator:

   $\lim_{\theta \to 0} 12 = 12$

Final answer:

$\boxed{12}$

Would you like further details or have any questions?

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Here are 5 related questions:

1. What is the general approximation of $\sin(x)$ and $\tan(x)$ for small $x$?
2. How does L'Hopital's rule apply to limits of indeterminate forms?
3. Can the limit $\lim_{\theta \to 0} \frac{\sin(5\theta) \cdot \tan(6\theta)}{\theta^2}$ be solved similarly?
4. What are higher-order approximations of $\sin(x)$ and $\tan(x)$?
5. What is the difference between $\lim_{\theta \to 0}$ and $\lim_{\theta \to \infty}$ in general?

Tip: When approaching limits involving small angles, approximations like $\sin(x) \approx x$ and $\tan(x) \approx x$ are very helpful for simplifying expressions.