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


{tan⁻¹(n)}

Verified step by step guidance

1. Recognize that the sequence is given by \( a_n = \tan^{-1}(n) \), where \( \tan^{-1} \) is the inverse tangent function, also known as arctangent.
2. Recall the behavior of the arctangent function: as its input \( x \) approaches infinity, \( \tan^{-1}(x) \) approaches a horizontal asymptote at \( \frac{\pi}{2} \).
3. Since \( n \) is a sequence that increases without bound (\( n \to \infty \)), consider the limit \( \lim_{n \to \infty} \tan^{-1}(n) \).
4. Use the known limit property: \( \lim_{x \to \infty} \tan^{-1}(x) = \frac{\pi}{2} \). This implies the sequence \( a_n = \tan^{-1}(n) \) converges to \( \frac{\pi}{2} \).
5. Conclude that the limit of the sequence \( \{ \tan^{-1}(n) \} \) as \( n \to \infty \) is \( \frac{\pi}{2} \).

Key Concepts

Limits of Sequences

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

Inverse Tangent Function (arctan)

The inverse tangent function, arctan(x), returns the angle whose tangent is x. It is defined for all real numbers and has horizontal asymptotes at ±π/2, meaning as x approaches ±∞, arctan(x) approaches ±π/2.

Behavior of arctan(n) as n → ∞

As the input n grows without bound, arctan(n) approaches its horizontal asymptote π/2. This means the sequence {arctan(n)} converges to π/2, since the values get closer and closer to this limit for large n.