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

{tan⁻¹(10n⁄(10n + 4))}

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Identify the general term of the sequence: \(a_n = \tan^{-1}\left(\frac{10n}{10n + 4}\right)\).

Analyze the behavior of the argument inside the inverse tangent function as \(n\) approaches infinity: consider the limit of \(\frac{10n}{10n + 4}\) as \(n \to \infty\).

Simplify the fraction \(\frac{10n}{10n + 4}\) by dividing numerator and denominator by \(n\), yielding \(\frac{10}{10 + \frac{4}{n}}\).

Evaluate the limit of the simplified fraction as \(n \to \infty\), noting that \(\frac{4}{n} \to 0\), so the fraction approaches \(\frac{10}{10} = 1\).

Use the continuity of the \(\tan^{-1}(x)\) function to conclude that the limit of the sequence is \(\tan^{-1}(1)\), which you can recognize or leave in this form.

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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.

Behavior of Rational Functions as n → ∞

For sequences involving rational expressions like 10n/(10n + 4), as n becomes very large, lower-order terms become negligible. The expression approaches the ratio of the leading coefficients, which helps simplify the limit evaluation.

Continuity and Limits of Inverse Trigonometric Functions

Inverse trigonometric functions like arctan are continuous, so the limit of arctan(f(n)) as n → ∞ equals arctan of the limit of f(n), provided the limit of f(n) exists. This property allows evaluating limits inside the function.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{tan⁻¹(10n⁄(10n + 4))}