Evaluate lim x approaches infinity arctan(x)

The limit of \(\arctan(x)\) as \(x\) approaches infinity can be evaluated as follows:

\[
\lim_{x \to \infty} \arctan(x)
\]

The function \(\arctan(x)\), also known as the inverse tangent function, has horizontal asymptotes as \(x\) approaches infinity and negative infinity. The asymptote as \(x \to \infty\) is \(\frac{\pi}{2}\), since:

- As \(x \to \infty\), \(\arctan(x)\) approaches but never exceeds \(\frac{\pi}{2}\).
- Similarly, as \(x \to -\infty\), \(\arctan(x)\) approaches \(-\frac{\pi}{2}\).

Thus, we have:

\[
\lim_{x \to \infty} \arctan(x) = \frac{\pi}{2}
\]

### Would you like further details or explanations on this topic?

Here are 5 related questions you might explore:
1. What is the limit of \(\arctan(x)\) as \(x \to -\infty\)?
2. How does the behavior of the arctangent function compare to other inverse trigonometric functions?
3. What are the applications of the arctangent function in real-world problems?
4. How do we derive the asymptotic behavior of \(\arctan(x)\) as \(x \to \infty\)?
5. What is the Taylor series expansion of \(\arctan(x)\) and how does it help in evaluating limits?

**Tip:** When dealing with limits of inverse trigonometric functions, remember their asymptotic behavior and corresponding horizontal asymptotes.

Explore the limit of the inverse trigonometric function arctan(x) as x approaches infinity. Understand the asymptotic behavior of arctangent and how it approaches \(\frac{\pi}{2}\). Suitable for advanced high school or early college students.

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