13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{n³⁄(n⁴ + 1)}

검증된 단계별 안내

Identify the given sequence: \(a_n = \frac{n^3}{n^4 + 1}\).

To find the limit as \(n\) approaches infinity, analyze the highest powers of \(n\) in the numerator and denominator.

Divide both the numerator and denominator by \(n^4\), the highest power in the denominator, to simplify the expression: \(a_n = \frac{\frac{n^3}{n^4}}{\frac{n^4}{n^4} + \frac{1}{n^4}} = \frac{\frac{1}{n}}{1 + \frac{1}{n^4}}\).

Evaluate the limit of each part as \(n \to \infty\): \(\frac{1}{n} \to 0\) and \(\frac{1}{n^4} \to 0\), so the expression simplifies to \(\frac{0}{1 + 0} = 0\).

Conclude that the limit of the sequence \(a_n\) as \(n\) approaches infinity is 0.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index n becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Dominant Term Analysis

When evaluating limits of sequences involving polynomials, focus on the highest degree terms in the numerator and denominator. These dominant terms determine the behavior of the sequence as n approaches infinity, simplifying the limit calculation.

Behavior of Rational Functions at Infinity

For sequences defined by rational functions, the limit as n approaches infinity depends on the degrees of the numerator and denominator polynomials. If the numerator's degree is less, the limit is zero; if equal, the limit is the ratio of leading coefficients; if greater, the sequence diverges.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{n³⁄(n⁴ + 1)}