Compute the limit of a_n = (3n^2 + 2n + 1)/(n^2 - 4) as n → ∞.

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Multiple Choice

Compute the limit of a_n = (3n^2 + 2n + 1)/(n^2 - 4) as n → ∞.

Explanation:
When looking at limits of rational polynomials as n grows large, the leading terms dictate the behavior. Here both numerator and denominator are like n^2, so divide top and bottom by n^2 to see the limit clearly: a_n = (3 + 2/n + 1/n^2) / (1 - 4/n^2). As n → ∞, the terms with 1/n and 1/n^2 vanish, leaving 3 in the numerator and 1 in the denominator. Therefore the limit is 3. This matches the idea that the ratio of leading coefficients is 3/1, and it rules out the other options since they would require different dominant behavior.

When looking at limits of rational polynomials as n grows large, the leading terms dictate the behavior. Here both numerator and denominator are like n^2, so divide top and bottom by n^2 to see the limit clearly: a_n = (3 + 2/n + 1/n^2) / (1 - 4/n^2). As n → ∞, the terms with 1/n and 1/n^2 vanish, leaving 3 in the numerator and 1 in the denominator. Therefore the limit is 3. This matches the idea that the ratio of leading coefficients is 3/1, and it rules out the other options since they would require different dominant behavior.

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