$$ \lim_{x\rarr\infty}\frac{x^2}{x^3}+\text{arctanx} $$

The given limit expression is:

\[
\lim_{x \to \infty} \left( \frac{x^2}{x^3} + \arctan(x) \right)
\]

Let's break this down:

1. **First term: $ \frac{x^2}{x^3} $:**

\[
\frac{x^2}{x^3} = \frac{1}{x}
\]

As $ x \to \infty $, $ \frac{1}{x} \to 0 $.

2. **Second term: $ \arctan(x) $:**

As $ x \to \infty $, $ \arctan(x) \to \frac{\pi}{2} $ because the arctangent function has a horizontal asymptote at $ \frac{\pi}{2} $.

### Combining both terms:
\[
\lim_{x \to \infty} \left( \frac{1}{x} + \arctan(x) \right) = 0 + \frac{\pi}{2} = \frac{\pi}{2}
\]

### Final Answer:
\[
\lim_{x \to \infty} \left( \frac{x^2}{x^3} + \arctan(x) \right) = \frac{\pi}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions to further explore:

1. What is the limit of $ \arctan(x) $ as $ x \to -\infty $?
2. How would the limit change if the power in the denominator were $ x^4 $ instead of $ x^3 $?
3. What are the horizontal asymptotes of the $ \arctan(x) $ function?
4. Can you evaluate the limit of $ \frac{x^3}{x^2} $ as $ x \to \infty $?
5. What happens if the arctangent term is replaced with $ \tan^{-1}(x^2) $?

**Tip:** Always check the behavior of individual terms as $ x \to \infty $ or $ x \to 0 $ to simplify complex limits.

Explore the limit of (x^2/x^3) + arctan(x) as x approaches infinity, breaking down the solution using the asymptotic behavior of trigonometric functions and limits. This problem is ideal for students in Grades 11-12 studying calculus and trigonometric limits.

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