Here are 20 calculus problems involving limits for you to solve.

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Basic Limits:

1. $\lim_{x \to 2} (3x + 5)$
2. $\lim_{x \to 0} \frac{x^2}{x + 1}$
3. $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
4. $\lim_{x \to \infty} \frac{5x + 3}{2x - 1}$
5. $\lim_{x \to 0} \sin(3x)$

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Intermediate Limits:

6. $\lim_{x \to 0} \frac{\tan(x)}{x}$
7. $\lim_{x \to \pi/4} \frac{\sin(x) - \cos(x)}{\cos(x)}$
8. $\lim_{x \to \infty} \sqrt{x^2 + 1} - x$
9. $\lim_{x \to -\infty} \frac{2x^3 + 5x}{x^2 - 7}$
10. $\lim_{x \to 0} \frac{e^x - 1}{x}$

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Challenging Limits:

11. $\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x}$
12. $\lim_{x \to \infty} \frac{\ln(x)}{x}$
13. $\lim_{x \to 0^+} x \ln(x)$
14. $\lim_{x \to \infty} \frac{x^2}{e^x}$
15. $\lim_{x \to 0} \frac{x^2}{\sin^2(x)}$

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Advanced Limits:

16. $\lim_{x \to 0} \frac{\sin(2x)}{\sin(3x)}$
17. $\lim_{x \to 0} \frac{\ln(1+x)}{x}$
18. $\lim_{x \to \infty} \frac{x^3}{e^{2x}}$
19. $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$
20. $\lim_{x \to 0^+} x^x$

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Would you like to start solving them, or shall I assist you with explanations? 😊

Expanding Your Understanding:

1. How does L'Hôpital's Rule simplify indeterminate forms like $\frac{0}{0}$?
2. Why does $\ln(x)$ grow slower than $x^n$ as $x \to \infty$?
3. What is the relationship between Taylor series and limits?
4. How can the Squeeze Theorem help evaluate certain limits?
5. Why do exponential terms dominate polynomial terms as $x \to \infty$?

Tip: Always check if a limit results in an indeterminate form before applying techniques like L'Hôpital's Rule.