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Give 30 similar problem to this negative or positive similarly 

I can help create 30 similar problems to the one in the image, using limits and functions that approach positive or negative values, involving rational expressions. 

### Problem in the image:
The limit you provided is:
\[
\lim_{{x \to 1}} \frac{x^3 - 1}{x - 1}
\]

Here are 30 similar problems, both positive and negative, involving limits and rational expressions:

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### Problems:

1. \(\lim_{{x \to 2}} \frac{x^3 - 8}{x - 2}\)
2. \(\lim_{{x \to 3}} \frac{x^3 - 27}{x - 3}\)
3. \(\lim_{{x \to -1}} \frac{x^3 + 1}{x + 1}\)
4. \(\lim_{{x \to 4}} \frac{x^3 - 64}{x - 4}\)
5. \(\lim_{{x \to 5}} \frac{x^3 - 125}{x - 5}\)
6. \(\lim_{{x \to -2}} \frac{x^3 + 8}{x + 2}\)
7. \(\lim_{{x \to 6}} \frac{x^3 - 216}{x - 6}\)
8. \(\lim_{{x \to 7}} \frac{x^3 - 343}{x - 7}\)
9. \(\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}\)
10. \(\lim_{{x \to 3}} \frac{x^2 - 9}{x - 3}\)
11. \(\lim_{{x \to 4}} \frac{x^2 - 16}{x - 4}\)
12. \(\lim_{{x \to 5}} \frac{x^2 - 25}{x - 5}\)
13. \(\lim_{{x \to -3}} \frac{x^2 - 9}{x + 3}\)
14. \(\lim_{{x \to -4}} \frac{x^2 - 16}{x + 4}\)
15. \(\lim_{{x \to -5}} \frac{x^2 - 25}{x + 5}\)
16. \(\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}\)
17. \(\lim_{{x \to 3}} \frac{x^2 - 9}{x - 3}\)
18. \(\lim_{{x \to 6}} \frac{x^3 - 216}{x - 6}\)
19. \(\lim_{{x \to 8}} \frac{x^2 - 64}{x - 8}\)
20. \(\lim_{{x \to -2}} \frac{x^3 + 8}{x + 2}\)
21. \(\lim_{{x \to -4}} \frac{x^3 + 64}{x + 4}\)
22. \(\lim_{{x \to -6}} \frac{x^3 + 216}{x + 6}\)
23. \(\lim_{{x \to 1}} \frac{2x^3 - 2}{x - 1}\)
24. \(\lim_{{x \to 2}} \frac{5x^2 - 20}{x - 2}\)
25. \(\lim_{{x \to 4}} \frac{3x^3 - 192}{x - 4}\)
26. \(\lim_{{x \to 1}} \frac{x^3 - 1}{x - 1}\)
27. \(\lim_{{x \to -3}} \frac{3x^2 + 9}{x + 3}\)
28. \(\lim_{{x \to 2}} \frac{(x - 2)^2}{x - 2}\)
29. \(\lim_{{x \to -5}} \frac{5x^2 + 25}{x + 5}\)
30. \(\lim_{{x \to 10}} \frac{x^2 - 100}{x - 10}\)

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### Tip:
When solving these limits, notice that many involve a difference of cubes or squares, and factoring can simplify the expression to eliminate the indeterminate form (e.g., \(0/0\)).

Let me know if you'd like to dive deeper into the solutions for any of these problems or if you have more questions!

Would you like further details on solving one or more of these problems?

Here are 5 related questions:
1. How do you factor a difference of cubes like \(x^3 - 1\)?
2. What is L'Hôpital's Rule and when is it applied?
3. What happens when a limit results in an indeterminate form like \(\frac{0}{0}\)?
4. How do you simplify rational expressions involving polynomials?
5. What role do conjugates play in simplifying limits with radicals?



Give 30 similar problem to this negative or positive similarly

This math problem involves finding the limit of a rational expression approaching an indeterminate form. We explore techniques such as factoring a difference of cubes to solve it. Explore 30 similar limit problems with both positive and negative cases, ideal for students in Grades 10-12.

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