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.