Math Problem Statement

Give 30 similar problem to this negative or positive similarly

Give 30 similar problem to this negative or positive similarly

Solution

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→1x3−1x−1\lim_{{x \to 1}} \frac{x^3 - 1}{x - 1}

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

Problems:

  1. lim⁡x→2x3−8x−2\lim_{{x \to 2}} \frac{x^3 - 8}{x - 2}
  2. lim⁡x→3x3−27x−3\lim_{{x \to 3}} \frac{x^3 - 27}{x - 3}
  3. lim⁡x→−1x3+1x+1\lim_{{x \to -1}} \frac{x^3 + 1}{x + 1}
  4. lim⁡x→4x3−64x−4\lim_{{x \to 4}} \frac{x^3 - 64}{x - 4}
  5. lim⁡x→5x3−125x−5\lim_{{x \to 5}} \frac{x^3 - 125}{x - 5}
  6. lim⁡x→−2x3+8x+2\lim_{{x \to -2}} \frac{x^3 + 8}{x + 2}
  7. lim⁡x→6x3−216x−6\lim_{{x \to 6}} \frac{x^3 - 216}{x - 6}
  8. lim⁡x→7x3−343x−7\lim_{{x \to 7}} \frac{x^3 - 343}{x - 7}
  9. lim⁡x→2x2−4x−2\lim_{{x \to 2}} \frac{x^2 - 4}{x - 2}
  10. lim⁡x→3x2−9x−3\lim_{{x \to 3}} \frac{x^2 - 9}{x - 3}
  11. lim⁡x→4x2−16x−4\lim_{{x \to 4}} \frac{x^2 - 16}{x - 4}
  12. lim⁡x→5x2−25x−5\lim_{{x \to 5}} \frac{x^2 - 25}{x - 5}
  13. lim⁡x→−3x2−9x+3\lim_{{x \to -3}} \frac{x^2 - 9}{x + 3}
  14. lim⁡x→−4x2−16x+4\lim_{{x \to -4}} \frac{x^2 - 16}{x + 4}
  15. lim⁡x→−5x2−25x+5\lim_{{x \to -5}} \frac{x^2 - 25}{x + 5}
  16. lim⁡x→1x2−1x−1\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}
  17. lim⁡x→3x2−9x−3\lim_{{x \to 3}} \frac{x^2 - 9}{x - 3}
  18. lim⁡x→6x3−216x−6\lim_{{x \to 6}} \frac{x^3 - 216}{x - 6}
  19. lim⁡x→8x2−64x−8\lim_{{x \to 8}} \frac{x^2 - 64}{x - 8}
  20. lim⁡x→−2x3+8x+2\lim_{{x \to -2}} \frac{x^3 + 8}{x + 2}
  21. lim⁡x→−4x3+64x+4\lim_{{x \to -4}} \frac{x^3 + 64}{x + 4}
  22. lim⁡x→−6x3+216x+6\lim_{{x \to -6}} \frac{x^3 + 216}{x + 6}
  23. lim⁡x→12x3−2x−1\lim_{{x \to 1}} \frac{2x^3 - 2}{x - 1}
  24. lim⁡x→25x2−20x−2\lim_{{x \to 2}} \frac{5x^2 - 20}{x - 2}
  25. lim⁡x→43x3−192x−4\lim_{{x \to 4}} \frac{3x^3 - 192}{x - 4}
  26. lim⁡x→1x3−1x−1\lim_{{x \to 1}} \frac{x^3 - 1}{x - 1}
  27. lim⁡x→−33x2+9x+3\lim_{{x \to -3}} \frac{3x^2 + 9}{x + 3}
  28. lim⁡x→2(x−2)2x−2\lim_{{x \to 2}} \frac{(x - 2)^2}{x - 2}
  29. lim⁡x→−55x2+25x+5\lim_{{x \to -5}} \frac{5x^2 + 25}{x + 5}
  30. lim⁡x→10x2−100x−10\lim_{{x \to 10}} \frac{x^2 - 100}{x - 10}

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., 0/00/0).

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 x3−1x^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 00\frac{0}{0}?
  4. How do you simplify rational expressions involving polynomials?
  5. What role do conjugates play in simplifying limits with radicals?

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Math Problem Analysis

Mathematical Concepts

Limits
Rational Expressions
Polynomials

Formulas

Limit of rational functions: lim(x→a) [f(x)/g(x)]
Factoring difference of cubes: x³ - a³ = (x - a)(x² + ax + a²)

Theorems

Factoring Theorem
Limit Properties
L'Hopital's Rule

Suitable Grade Level

Grades 10-12

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