Math Problem Statement

Hitunglah nilai limit berikut: lim x -> ∞ (3x^2 - 2)^2 / (4x^3 + 3x^2 - 2)

Solution

The limit problem you've uploaded is:

limx(3x22)24x3+3x22\lim_{{x \to \infty}} \frac{(3x^2 - 2)^2}{4x^3 + 3x^2 - 2}

Step-by-Step Solution:

  1. Expand the numerator: (3x22)2=(3x2)223x22+(2)2=9x412x2+4(3x^2 - 2)^2 = (3x^2)^2 - 2 \cdot 3x^2 \cdot 2 + (-2)^2 = 9x^4 - 12x^2 + 4 So, the expression becomes: limx9x412x2+44x3+3x22\lim_{{x \to \infty}} \frac{9x^4 - 12x^2 + 4}{4x^3 + 3x^2 - 2}

  2. Factor out the highest powers of xx: In the numerator, the highest degree term is 9x49x^4, and in the denominator, the highest degree term is 4x34x^3.

    Divide both the numerator and the denominator by x3x^3, the highest degree in the denominator:

    limxx4(912x2+4x4)x3(4+3x2x3)\lim_{{x \to \infty}} \frac{x^4(9 - \frac{12}{x^2} + \frac{4}{x^4})}{x^3(4 + \frac{3}{x} - \frac{2}{x^3})}

    Simplifying the x4/x3=xx^4/x^3 = x, we get:

    limxx912x2+4x44+3x2x3\lim_{{x \to \infty}} x \cdot \frac{9 - \frac{12}{x^2} + \frac{4}{x^4}}{4 + \frac{3}{x} - \frac{2}{x^3}}

  3. Evaluate the limit as xx \to \infty: As xx \to \infty, the terms involving 1xn\frac{1}{x^n} approach zero. So we are left with:

    x94x \cdot \frac{9}{4}

    As xx \to \infty, this expression grows without bound. Therefore, the limit is:

    limx(3x22)24x3+3x22=\lim_{{x \to \infty}} \frac{(3x^2 - 2)^2}{4x^3 + 3x^2 - 2} = \infty

Conclusion:

The limit as xx \to \infty is \infty.


Do you want further details or have any questions about this problem?

Here are five related questions to explore further:

  1. How would the result change if the highest power of xx in the denominator was higher than that in the numerator?
  2. Can this limit approach a finite value for different functions?
  3. What happens to limits if we introduce logarithmic functions in the numerator or denominator?
  4. How does L'Hôpital's Rule apply in cases where both numerator and denominator approach infinity?
  5. How does polynomial degree comparison help in finding limits at infinity?

Tip: When dealing with limits approaching infinity, always look at the highest degree terms in both the numerator and denominator for a quick estimate of the behavior.

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Math Problem Analysis

Mathematical Concepts

Limits
Algebraic expressions
Asymptotic behavior

Formulas

Limit definition: lim x -> ∞
Polynomial long division

Theorems

Degree comparison for limits at infinity
Limits of rational functions

Suitable Grade Level

Grades 10-12