Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
15. lim(x→∞)arctan(x)

Verified step by step guidance

1. Recall the definition and behavior of the arctangent function, \(\arctan(x)\), which is the inverse of the tangent function restricted to \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
2. Understand that as \(x\) approaches infinity, the value of \(\arctan(x)\) approaches a horizontal asymptote because the tangent function has vertical asymptotes at \(\pm \frac{\pi}{2}\).
3. Recognize that \(\arctan(x)\) is an increasing function and its range is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), so the limit as \(x \to \infty\) must be the upper bound of this range.
4. Express the limit formally as \(\lim_{x \to \infty} \arctan(x) = L\), where \(L\) is the horizontal asymptote value to be identified.
5. Conclude that the limit is the horizontal asymptote \(L = \frac{\pi}{2}\), since \(\arctan(x)\) approaches this value as \(x\) becomes very large.

Key Concepts

Limit of a Function at Infinity

The limit of a function as x approaches infinity describes the value that the function approaches as x grows larger without bound. It helps determine the end behavior of the function, which is essential for understanding asymptotic behavior and horizontal asymptotes.

Arctangent Function Properties

The arctangent function, denoted arctan(x), is the inverse of the tangent function restricted to its principal domain. It is continuous and increasing, with horizontal asymptotes at ±π/2, meaning as x approaches infinity, arctan(x) approaches π/2.

Using Graphs to Understand Limits

Graphs provide a visual representation of a function’s behavior, making it easier to estimate limits, especially at infinity. Observing the graph of arctan(x) shows how the function approaches its horizontal asymptote, reinforcing the analytical limit result.