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

{(√(4n⁴ + 3n))⁄(8n² + 1)}

검증된 단계별 안내

Identify the given sequence: \(a_n = \frac{\sqrt{4n^4 + 3n}}{8n^2 + 1}\).

To find the limit as \(n \to \infty\), first analyze the dominant terms in the numerator and denominator. The highest power of \(n\) inside the square root is \(n^4\), and in the denominator it is \(n^2\).

Rewrite the numerator by factoring out \(n^4\) inside the square root: \(\sqrt{4n^4 + 3n} = \sqrt{n^4(4 + \frac{3}{n^3})} = n^2 \sqrt{4 + \frac{3}{n^3}}\).

Rewrite the denominator as \(8n^2 + 1 = n^2(8 + \frac{1}{n^2})\).

Express the sequence as \(a_n = \frac{n^2 \sqrt{4 + \frac{3}{n^3}}}{n^2 (8 + \frac{1}{n^2})} = \frac{\sqrt{4 + \frac{3}{n^3}}}{8 + \frac{1}{n^2}}\). Then, take the limit as \(n \to \infty\) by evaluating the limits of the numerator and denominator separately.

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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 goes to infinity. If the terms get arbitrarily close to a finite number, the sequence converges; otherwise, it diverges. Understanding this helps determine the behavior of sequences for large n.

Asymptotic Behavior and Dominant Terms

When evaluating limits of sequences involving polynomials or roots, focus on the highest-degree terms in numerator and denominator. These dominant terms dictate the growth rate and simplify the expression, making it easier to find the limit as n approaches infinity.

Simplification Using Algebraic Manipulation

Techniques like factoring, dividing numerator and denominator by the highest power of n, or rationalizing expressions help simplify complex sequences. This process reveals the underlying behavior of the sequence and aids in accurately computing the limit.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{(√(4n⁴ + 3n))⁄(8n² + 1)}