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To analyze the limit of the given sequence as \( n \) tends to infinity, let's break down the expression.

Given sequence:
\[
a_n = \arctan(3n)
\]

### Step 1: Understand the Behavior of the Sequence
The function \(\arctan(x)\) (inverse tangent) is defined for all real numbers \(x\) and it has the following key properties:
- As \( x \) approaches \( \infty \), \(\arctan(x)\) approaches \( \frac{\pi}{2} \).
- As \( x \) approaches \( -\infty \), \(\arctan(x)\) approaches \( -\frac{\pi}{2} \).

### Step 2: Apply the Limit
We need to find:
\[
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \arctan(3n)
\]

As \( n \) tends to infinity, \( 3n \) also tends to infinity. Therefore:
\[
\lim_{n \to \infty} \arctan(3n) = \frac{\pi}{2}
\]

### Conclusion
The limit of the sequence \( a_n = \arctan(3n) \) as \( n \) tends to infinity is \( \frac{\pi}{2} \).

Would you like more details on any specific part of the calculation, or do you have any further questions?

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Here are 5 related questions to expand your understanding:

1. What is the behavior of \(\arctan(x)\) as \(x\) approaches zero?
2. How would the limit change if the sequence were \(a_n = \arctan\left(\frac{3n}{n+1}\right)\)?
3. What is the significance of the horizontal asymptotes in the context of the \(\arctan(x)\) function?
4. How would you calculate the limit if \(a_n = \arctan\left(\frac{3n}{n^2}\right)\)?
5. What is the limit of the sequence \(a_n = \arctan\left(\frac{1}{3n}\right)\) as \(n\) approaches infinity?

**Tip:** When dealing with limits that involve trigonometric functions, especially those like \(\arctan(x)\), always consider the behavior of the function as the argument approaches infinity or zero. This will often simplify the process significantly.

Explore the limit of the sequence arctan(3n) as n tends to infinity with a step-by-step analysis. Understand the behavior of trigonometric functions and their limits. Suitable for students in Grades 10-12.

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