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The image contains a series of limit problems that involve trigonometric functions as \( x \to 0 \). Let’s solve these limits one by one.

### 1. \( \lim_{x \to 0} \frac{\tan^2 2x}{2x^2} \)
   Using the approximation \( \tan x \approx x \) as \( x \to 0 \), we get:
   \[
   \lim_{x \to 0} \frac{\tan^2(2x)}{2x^2} = \lim_{x \to 0} \frac{(2x)^2}{2x^2} = 4.
   \]
   So, the value is **4**.

### 2. \( \lim_{x \to 0} \frac{\sin 6x}{x \cos 2x} \)
   We use the approximations \( \sin x \approx x \) and \( \cos x \approx 1 \) as \( x \to 0 \):
   \[
   \lim_{x \to 0} \frac{\sin(6x)}{x \cos(2x)} = \lim_{x \to 0} \frac{6x}{x \cdot 1} = 6.
   \]
   So, the value is **6**.

### 3. \( \lim_{x \to 0} \frac{\sin 2x}{\tan 8x} \)
   Using \( \sin x \approx x \) and \( \tan x \approx x \) as \( x \to 0 \):
   \[
   \lim_{x \to 0} \frac{\sin(2x)}{\tan(8x)} = \lim_{x \to 0} \frac{2x}{8x} = \frac{2}{8} = \frac{1}{4}.
   \]
   So, the value is **\(\frac{1}{4}\)**.

### 4. \( \lim_{x \to 0} \frac{x}{\sin x} \)
   This is a standard limit that can be directly computed using the approximation \( \sin x \approx x \) as \( x \to 0 \):
   \[
   \lim_{x \to 0} \frac{x}{\sin x} = 1.
   \]
   So, the value is **1**.

### 5. \( \lim_{x \to 0} \frac{\tan 3x}{5x} \)
   Using \( \tan x \approx x \) as \( x \to 0 \):
   \[
   \lim_{x \to 0} \frac{\tan(3x)}{5x} = \lim_{x \to 0} \frac{3x}{5x} = \frac{3}{5}.
   \]
   So, the value is **\(\frac{3}{5}\)**.

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Would you like any further details or explanations on these solutions? 

Here are some related questions to consider:
1. How do approximations like \( \sin x \approx x \) help in limit problems?
2. What is the general approach to solving limits involving trigonometric functions?
3. Can you explain why \( \lim_{x \to 0} \frac{x}{\sin x} = 1 \) is important?
4. What happens if the limits involve higher powers of \( x \) or trigonometric terms?
5. How would the limits change if \( x \) were approaching values other than 0?

**Tip:** Always remember that trigonometric approximations \( \sin x \approx x \) and \( \tan x \approx x \) are very useful for limit problems as \( x \to 0 \).

Calculate the limits of trigonometric expressions as x approaches 0.

Explore the solutions to limit problems involving trigonometric functions such as sin(x) and tan(x) as x approaches 0. Learn key concepts, formulas, and theorems with step-by-step explanations. Suitable for students in Grades 11-12.

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