What is the limit as x approaches infinity of (2x^3 + x^2)/(5x^3 - 2)?

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Multiple Choice

What is the limit as x approaches infinity of (2x^3 + x^2)/(5x^3 - 2)?

Explanation:
As x grows without bound, the terms with the highest power of x dominate in a rational function. Here both the numerator and the denominator are cubic polynomials, so the limit is determined by the leading coefficients. The leading term in the numerator is 2x^3 and in the denominator is 5x^3, giving 2/5. Dividing both numerator and denominator by x^3 confirms this: (2 + 1/x) / (5 - 2/x^3) → 2/5 as x → ∞.

As x grows without bound, the terms with the highest power of x dominate in a rational function. Here both the numerator and the denominator are cubic polynomials, so the limit is determined by the leading coefficients. The leading term in the numerator is 2x^3 and in the denominator is 5x^3, giving 2/5. Dividing both numerator and denominator by x^3 confirms this: (2 + 1/x) / (5 - 2/x^3) → 2/5 as x → ∞.

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