https://file.aiquickdraw.com/mathbot/file/1727478823132heisew8w.jpg

Let's work through the requested limit problems from number 8 to 14 step by step.

### Problem 8:
\[
\lim_{\theta \to 0} \frac{\sin 2\theta}{\sin \theta}
\]
Using the double-angle identity for sine, \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\), we rewrite the limit:
\[
\lim_{\theta \to 0} \frac{2\sin(\theta)\cos(\theta)}{\sin(\theta)}
\]
Simplifying:
\[
\lim_{\theta \to 0} 2\cos(\theta)
\]
Since \(\cos(0) = 1\), the limit is:
\[
\boxed{2}
\]

---

### Problem 9:
\[
\lim_{\theta \to 0} \frac{\cot(\pi \theta) \sin(\theta)}{2\sec(\theta)}
\]
We know that \(\cot(x) = \frac{\cos(x)}{\sin(x)}\) and \(\sec(x) = \frac{1}{\cos(x)}\), so substitute these into the limit:
\[
\lim_{\theta \to 0} \frac{\frac{\cos(\pi \theta)}{\sin(\pi \theta)} \sin(\theta)}{2\frac{1}{\cos(\theta)}}
\]
Simplifying:
\[
\lim_{\theta \to 0} \frac{\cos(\pi \theta)\sin(\theta)\cos(\theta)}{2\sin(\pi \theta)}
\]
As \(\theta \to 0\), both \(\sin(\theta)\) and \(\sin(\pi \theta)\) tend to 0, but using L'Hopital's rule or small-angle approximations for sine and cosine, we can find:
\[
\boxed{0}
\]

---

### Problem 10:
\[
\lim_{t \to 0} \frac{\sin^2(3t)}{2t}
\]
First, use the identity \(\sin^2(x) = (1 - \cos(2x))/2\) and substitute, or alternatively, factor the expression as follows:
\[
\lim_{t \to 0} \frac{\sin(3t)}{t} \cdot \frac{\sin(3t)}{2}
\]
Using \(\lim_{t \to 0} \frac{\sin(3t)}{t} = 3\), the limit becomes:
\[
3 \cdot 0 = \boxed{0}
\]

---

### Problem 11:
\[
\lim_{t \to 0} \frac{\tan^2(3t)}{2t}
\]
We can use the approximation \(\tan(x) \approx x\) when \(x\) is small. Thus:
\[
\lim_{t \to 0} \frac{(3t)^2}{2t} = \lim_{t \to 0} \frac{9t^2}{2t} = \lim_{t \to 0} \frac{9t}{2} = 0
\]
Thus, the limit is:
\[
\boxed{0}
\]

---

### Problem 12:
\[
\lim_{\theta \to 0} \frac{\tan(2\theta)}{\sin(2\theta) - 1}
\]
Using \(\tan(x) \approx x\) and \(\sin(x) \approx x\) for small angles:
\[
\lim_{\theta \to 0} \frac{2\theta}{2\theta - 1}
\]
At \(\theta = 0\), the denominator becomes \(-1\), so the limit is:
\[
\boxed{0}
\]

---

### Problem 13:
\[
\lim_{t \to 0} \frac{\sin(3t) + 4t}{t \sec(t)}
\]
We use the small-angle approximation for sine and secant (\(\sec(t) \approx 1 + t^2/2\)):
\[
\lim_{t \to 0} \frac{3t + 4t}{t(1)} = \lim_{t \to 0} \frac{7t}{t} = 7
\]
Thus, the limit is:
\[
\boxed{7}
\]

---

### Problem 14:
\[
\lim_{\theta \to 0} \frac{\sin^2(\theta)}{\theta^2}
\]
We recognize this as a standard limit form. Since \(\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1\), we have:
\[
\lim_{\theta \to 0} \frac{\sin^2(\theta)}{\theta^2} = \left( \lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} \right)^2 = 1^2 = 1
\]
Thus, the limit is:
\[
\boxed{1}
\]

---

Would you like further details or have any questions about the steps?

Here are 5 related questions to explore:
1. What is L'Hopital's Rule and how can it be applied to limits?
2. How do approximations for small angles (\(\sin(x) \approx x\), \(\tan(x) \approx x\)) help simplify limit problems?
3. What is the significance of the standard limit \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\)?
4. How do trigonometric identities assist in solving complex limit problems?
5. When do we need to use Taylor series expansions in evaluating limits?

**Tip:** When evaluating limits involving trigonometric functions, always check if small-angle approximations can simplify the expression!

This guide covers step-by-step solutions for trigonometric limit problems, including problems involving small-angle approximations, trigonometric identities, and L'Hopital's Rule. Focused on problems 8-14.

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