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

{nsin(6 / n)}

검증된 단계별 안내

Identify the sequence given: \(a_n = n \sin\left(\frac{6}{n}\right)\).

Recall the limit property for sine near zero: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).

Rewrite the sequence term to use this property by setting \(x = \frac{6}{n}\), so \(a_n = n \sin\left(\frac{6}{n}\right) = \frac{\sin\left(\frac{6}{n}\right)}{\frac{6}{n}} \times 6\).

Analyze the limit as \(n \to \infty\): since \(\frac{6}{n} \to 0\), use the limit property to find \(\lim_{n \to \infty} \frac{\sin\left(\frac{6}{n}\right)}{\frac{6}{n}} = 1\).

Combine the results to conclude that \(\lim_{n \to \infty} a_n = 6 \times 1 = 6\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Limit of a Sequence

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.

Behavior of the Sine Function for Small Arguments

For values of x close to zero, sin(x) is approximately equal to x. This linear approximation, sin(x) ≈ x, is useful for evaluating limits involving sine functions where the argument tends to zero.

Limit Laws and Substitution

Limit laws allow the evaluation of limits by breaking complex expressions into simpler parts. Substitution involves replacing variables with their limiting values when the function is continuous, facilitating the calculation of the sequence's limit.

Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{nsin(6 / n)}