Math Problem Statement

Compute the limit lim(x → -∞) ((-6x - 4) / (8x^3 - 6x^2 + x + 4))

Solution

To solve the given limit problem:

limx6x48x36x2+x+4\lim_{{x \to -\infty}} \frac{-6x - 4}{8x^3 - 6x^2 + x + 4}

Step-by-step process:

  1. Examine the degrees of the polynomial:

    • In the numerator: 6x4-6x - 4 has a degree of 1.
    • In the denominator: 8x36x2+x+48x^3 - 6x^2 + x + 4 has a degree of 3.
  2. Leading terms for large xx:

    • The leading term in the numerator is 6x-6x.
    • The leading term in the denominator is 8x38x^3.
  3. Simplify by dividing numerator and denominator by x3x^3 (the highest power of xx in the denominator): 6x48x36x2+x+4=6x4x38x36x2+x+4x3\frac{-6x - 4}{8x^3 - 6x^2 + x + 4} = \frac{\frac{-6x - 4}{x^3}}{\frac{8x^3 - 6x^2 + x + 4}{x^3}}

  4. Simplify each term:

    • Numerator: 6x4x3=6x24x3\frac{-6x - 4}{x^3} = \frac{-6}{x^2} - \frac{4}{x^3}.
    • Denominator: 8x3x36x2x3+xx3+4x3=86x+1x2+4x3\frac{8x^3}{x^3} - \frac{6x^2}{x^3} + \frac{x}{x^3} + \frac{4}{x^3} = 8 - \frac{6}{x} + \frac{1}{x^2} + \frac{4}{x^3}.
  5. Take the limit as xx \to -\infty:

    • The terms involving x1x^{-1}, x2x^{-2}, and x3x^{-3} in both the numerator and denominator will approach 0 as xx \to -\infty.
    • Thus, the numerator approaches 0.
    • The denominator approaches 88.
  6. Conclusion: The limit is: limx08=0\lim_{{x \to -\infty}} \frac{0}{8} = 0

Therefore, the correct answer is 0.

Would you like more details on any step, or do you have any questions?

Related Questions:

  1. How do you determine which term dominates in the limit at infinity?
  2. What would happen if the degrees of the numerator and denominator were the same?
  3. How do you handle limits approaching infinity versus negative infinity?
  4. What happens when the numerator has a higher degree than the denominator?
  5. Can we use L'Hopital's rule to solve this limit?

Tip: When solving limits involving polynomials, always look for the highest degree terms, as they dominate the behavior at large values of xx.

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Math Problem Analysis

Mathematical Concepts

Limits at Infinity
Polynomials

Formulas

Leading term comparison of polynomials

Theorems

Limit of polynomials as x approaches infinity

Suitable Grade Level

Grades 11-12 or College Level