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The given problem asks for the limit:

\[
\lim_{\theta \to 0} \frac{\cos(\theta) - 1}{\sin(\theta)}
\]

### Solution:

Direct substitution of \(\theta = 0\) leads to an indeterminate form of \(\frac{0}{0}\), so we need to simplify it.

#### Method 1: Using L'Hôpital's Rule

L'Hôpital's Rule is applicable when we encounter indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). The rule states that:

\[
\lim_{\theta \to 0} \frac{f(\theta)}{g(\theta)} = \lim_{\theta \to 0} \frac{f'(\theta)}{g'(\theta)}
\]

provided the right-hand limit exists.

1. The numerator \(f(\theta) = \cos(\theta) - 1\), and its derivative is:
   \[
   f'(\theta) = \frac{d}{d\theta} [\cos(\theta) - 1] = -\sin(\theta)
   \]

2. The denominator \(g(\theta) = \sin(\theta)\), and its derivative is:
   \[
   g'(\theta) = \frac{d}{d\theta} [\sin(\theta)] = \cos(\theta)
   \]

Now, applying L'Hôpital's Rule:
\[
\lim_{\theta \to 0} \frac{\cos(\theta) - 1}{\sin(\theta)} = \lim_{\theta \to 0} \frac{-\sin(\theta)}{\cos(\theta)}
\]

Substituting \(\theta = 0\):
\[
\frac{-\sin(0)}{\cos(0)} = \frac{0}{1} = 0
\]

Thus, the limit is:

\[
\boxed{0}
\]

Would you like more details or have any questions on this process? Here are five related questions you might explore:

1. How can limits involving trigonometric functions be solved without L'Hôpital's Rule?
2. What is L'Hôpital's Rule and when is it applied?
3. What are other techniques for finding limits besides L'Hôpital's Rule?
4. How does the Taylor series expansion of trigonometric functions help in solving limits?
5. Can limits involving indeterminate forms like \(\frac{0}{0}\) always be resolved with L'Hôpital's Rule?

**Tip:** Always check if the limit can be simplified algebraically before applying L'Hôpital's Rule, as it may not be necessary in simpler cases.

Find the limit: lim(θ→0) [(cos(θ) - 1) / sin(θ)]

Learn how to find the limit of (cos(θ) - 1) / sin(θ) as θ approaches 0 using L'Hôpital's Rule. This problem explores key trigonometric concepts and indeterminate forms in calculus. Suitable for college or AP Calculus students.

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