https://file.aiquickdraw.com/mathbot/file/17270582908551ii1acvg.jpg

The limit you are asked to find is:

\[
\lim_{\theta \to 0} \theta \cos \theta
\]

You are provided with the known limit:

\[
\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1.
\]

### Solution:
We can solve this limit directly.

\[
\lim_{\theta \to 0} \theta \cos \theta = \lim_{\theta \to 0} \theta \cdot \lim_{\theta \to 0} \cos \theta.
\]

1. First, evaluate \(\lim_{\theta \to 0} \theta\). Since \(\theta\) approaches 0, this part will become 0:
   \[
   \lim_{\theta \to 0} \theta = 0.
   \]

2. Next, evaluate \(\lim_{\theta \to 0} \cos \theta\). Since \(\cos \theta\) approaches 1 as \(\theta\) approaches 0:
   \[
   \lim_{\theta \to 0} \cos \theta = 1.
   \]

3. Finally, combine the two limits:
   \[
   \lim_{\theta \to 0} \theta \cos \theta = 0 \cdot 1 = 0.
   \]

### Final Answer:
\[
\lim_{\theta \to 0} \theta \cos \theta = 0.
\]

Would you like more details or further clarifications on this? 

Here are 5 related questions:
1. How does \(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\) help in solving other trigonometric limits?
2. What are some other common trigonometric limits that are useful in calculus?
3. How would you approach limits that involve \(\tan \theta\) instead of \(\sin \theta\)?
4. How do you handle limits where both numerator and denominator approach zero (indeterminate forms)?
5. Can L'Hopital's Rule be applied to this problem, and if so, how?

**Tip:** For trigonometric limits, knowing the small angle approximations for \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) can simplify many limit problems!

Find the limit lim(θ → 0) θ cos(θ) using lim(θ → 0) (sin(θ)/θ) = 1.

Solve the limit lim(θ → 0) θ cos(θ) using the known limit lim(θ → 0) (sin(θ)/θ) = 1. This trigonometric limit problem is suitable for high school calculus students. Learn how to break down the components using standard limit theorems.

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