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


{(3n³ − 1)⁄(2n³ + 1)}

Verified step by step guidance

1. Identify the given sequence: \(a_n = \frac{3n^3 - 1}{2n^3 + 1}\).
2. To find the limit as \(n\) approaches infinity, observe the highest power of \(n\) in the numerator and denominator, which is \(n^3\).
3. Divide both the numerator and the denominator by \(n^3\) to simplify the expression: \(\frac{\frac{3n^3}{n^3} - \frac{1}{n^3}}{\frac{2n^3}{n^3} + \frac{1}{n^3}} = \frac{3 - \frac{1}{n^3}}{2 + \frac{1}{n^3}}\).
4. As \(n\) approaches infinity, the terms \(\frac{1}{n^3}\) approach 0, so the expression simplifies to \(\frac{3 - 0}{2 + 0} = \frac{3}{2}\).
5. Conclude that the limit of the sequence is \(\frac{3}{2}\).

Key Concepts

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.