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To determine the limit \(\lim_{\theta \to 0} \frac{1 - \cos \theta}{\sin 4\theta}\), we can use the standard trigonometric limit \(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\).

### Step-by-Step Solution:

1. **Rewrite the expression using identities**:
   The expression can be rewritten using the trigonometric identity:
   \[
   1 - \cos \theta = 2\sin^2\left(\frac{\theta}{2}\right)
   \]
   Therefore, the limit becomes:
   \[
   \lim_{\theta \to 0} \frac{2\sin^2\left(\frac{\theta}{2}\right)}{\sin 4\theta}
   \]

2. **Simplify the limit**:
   We can break down the expression into a product of two limits:
   \[
   \lim_{\theta \to 0} \frac{2\sin^2\left(\frac{\theta}{2}\right)}{\sin 4\theta} = 2 \cdot \lim_{\theta \to 0} \frac{\sin^2\left(\frac{\theta}{2}\right)}{\theta^2} \cdot \lim_{\theta \to 0} \frac{\theta^2}{\sin 4\theta}
   \]

3. **Apply the trigonometric limit**:
   First, use the limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\):
   \[
   \lim_{\theta \to 0} \frac{\sin\left(\frac{\theta}{2}\right)}{\frac{\theta}{2}} = 1
   \]
   Hence,
   \[
   \lim_{\theta \to 0} \frac{\sin^2\left(\frac{\theta}{2}\right)}{\theta^2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
   \]
   Now, for the second limit:
   \[
   \lim_{\theta \to 0} \frac{\theta^2}{\sin 4\theta} = \lim_{\theta \to 0} \frac{\theta^2}{4\theta} \cdot \frac{4\theta}{\sin 4\theta} = \frac{1}{4}
   \]

4. **Combine the results**:
   Therefore, the overall limit is:
   \[
   2 \cdot \frac{1}{4} \cdot \frac{1}{4} = \frac{1}{8}
   \]

Thus, the limit is:
\[
\lim_{\theta \to 0} \frac{1 - \cos \theta}{\sin 4\theta} = \frac{1}{8}
\]

Would you like more details or have any questions?

### Relative Questions:
1. What other trigonometric identities can be useful in solving limits?
2. How can you apply L'Hôpital's Rule to solve similar trigonometric limits?
3. Why is \(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\) fundamental in calculus?
4. What are the steps to solve limits involving complex trigonometric functions?
5. How can you verify the limit \(\lim_{\theta \to 0} \frac{1 - \cos \theta}{\sin 4\theta} = \frac{1}{8}\) using a different method?

### Tip:
When dealing with trigonometric limits, always consider if you can use known limits and identities to simplify the expression before applying more complex techniques like L'Hôpital's Rule.

Use the relation lim (θ → 0) (sin θ / θ) = 1 to determine the limit: lim (θ → 0) (1 - cos θ) / sin 4θ

Learn how to calculate the limit lim (θ → 0) (1 - cos θ) / sin 4θ using the trigonometric identity 1 - cos θ = 2sin²(θ/2) and the fundamental trigonometric limit lim (θ → 0) (sin θ / θ) = 1. This problem is suitable for students in Grades 11-12 and covers key calculus concepts.

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