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The given limit is:

\[
\lim_{\theta \to 0} \frac{\theta + \theta \cos(3\theta)}{\sin(90^\circ) \cos(4\theta)}
\]

### Step-by-Step Solution:

1. **Simplify the Trigonometric Terms**:
   - \(\sin(90^\circ) = 1\) (since \(\sin(90^\circ)\) is a standard trigonometric value).
   - So, the limit becomes:
     \[
     \lim_{\theta \to 0} \frac{\theta + \theta \cos(3\theta)}{\cos(4\theta)}
     \]

2. **Factor out \(\theta\) in the numerator**:
   \[
   \lim_{\theta \to 0} \frac{\theta(1 + \cos(3\theta))}{\cos(4\theta)}
   \]

3. **Evaluate the limit as \(\theta \to 0\)**:
   - \(\cos(3\theta) \to \cos(0) = 1\)
   - \(\cos(4\theta) \to \cos(0) = 1\)

Thus, the expression simplifies to:
\[
\lim_{\theta \to 0} \frac{\theta(1 + 1)}{1} = \lim_{\theta \to 0} 2\theta = 0
\]

### Final Answer:
\[
\lim_{\theta \to 0} \frac{\theta + \theta \cos(3\theta)}{\sin(90^\circ) \cos(4\theta)} = 0
\]

Would you like more details on this solution? Here are some related questions:

1. What are the standard limits for trigonometric functions as \(\theta\) approaches 0?
2. How does factoring help in solving limits?
3. Why does \(\cos(0) = 1\) and how does that simplify trigonometric expressions?
4. How does L'Hopital's rule apply to limits like these?
5. What happens if the denominator tends to 0 in such limit problems?

**Tip**: Always evaluate the behavior of trigonometric functions at the limit points to simplify expressions quickly.

lim(θ → 0) (θ + θ cos(3θ)) / (sin(90°) cos(4θ))

This problem involves evaluating the limit of a trigonometric expression as θ approaches 0. Learn how to simplify the trigonometric functions using standard limit laws, resulting in a final answer of 0. This solution is ideal for university-level calculus students.

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