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The image contains two sets of limit problems:

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**429. Limits as \( x \to 0 \):**

a) \( \lim_{x \to 0} \frac{\sin 7x - \sin 3x}{5x} \)

b) \( \lim_{x \to 0} \frac{1 - \cos 4x}{x \cdot \tan 2x} \)

c) \( \lim_{x \to 0} \frac{1 - \sqrt{\cos 2x}}{x \cdot \arcsin 7x} \)

d) \( \lim_{x \to 0} \frac{\cos 5x - \cos 9x}{3x \cdot \tan 5x} \)

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**430. Limits as \( x \to \infty \) or \( x \to 0 \):**

a) \( \lim_{x \to +\infty} \left( \frac{2x - 1}{2x + 1} \right)^{2x + 3} \)

b) \( \lim_{x \to +\infty} \left( \frac{3x + 2}{3x - 1} \right)^{2x - 1} \)

c) \( \lim_{x \to 0} \frac{\ln(x^2 + 1)}{\sqrt{x^2 + 1} - 1} \)

d) \( \lim_{x \to 1} \frac{x^2 - 1}{\ln x} \)

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Would you like to solve one or more specific limits from this list? Let me know if you'd like any clarifications.

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Here are 5 questions for deeper exploration:
1. What is the geometric interpretation of the sine and cosine limit forms?
2. How can L'Hopital's rule be applied to solve limits of indeterminate forms?
3. What are Taylor expansions, and how can they help in solving limits involving trigonometric functions?
4. How do logarithmic limits differ in behavior near singularities (e.g., at 0 or \( \infty \))?
5. What are common strategies for solving limits involving complex expressions (e.g., combining algebraic simplifications with limits)?

**Tip:** In limit problems, always consider the possibility of indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), and use strategies like L'Hopital's rule or series expansions to resolve them.

The image contains two sets of limit problems: 429 and 430.

Solve limit problems involving trigonometric and logarithmic functions with step-by-step explanations. The problems cover concepts like limits of trigonometric expressions, L'Hopital's rule, and logarithmic limits. Suitable for high school or college-level calculus students.

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