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

{(3n³ − 1)⁄(2n³ + 1)}

검증된 단계별 안내

Identify the given sequence: \(a_n = \frac{3n^3 - 1}{2n^3 + 1}\).

To find the limit as \(n\) approaches infinity, observe the highest power of \(n\) in the numerator and denominator, which is \(n^3\).

Divide both the numerator and the denominator by \(n^3\) to simplify the expression: \(\frac{\frac{3n^3}{n^3} - \frac{1}{n^3}}{\frac{2n^3}{n^3} + \frac{1}{n^3}} = \frac{3 - \frac{1}{n^3}}{2 + \frac{1}{n^3}}\).

As \(n\) approaches infinity, the terms \(\frac{1}{n^3}\) approach 0, so the expression simplifies to \(\frac{3 - 0}{2 + 0} = \frac{3}{2}\).

Conclude that the limit of the sequence is \(\frac{3}{2}\).

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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 is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Dominant Term in Polynomial Expressions

When evaluating limits of sequences involving polynomials, the highest degree terms dominate the behavior as n becomes large. Lower degree terms become insignificant, so the limit can often be found by comparing the leading terms.

Limit Laws and Simplification Techniques

Limit laws allow the separation and simplification of complex expressions. For rational sequences, dividing numerator and denominator by the highest power of n helps simplify the expression and find the limit more easily.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{(3n³ − 1)⁄(2n³ + 1)}