We are tasked with finding the limit of the following expression as \( n \to \infty \):

\[
\lim_{n \to \infty} \frac{n+2}{n^2 + 1}.
\]

### Step-by-Step Solution:

1. **Expression Review**:
   The expression is \( \frac{n+2}{n^2 + 1} \). As \( n \) grows large, both the numerator and the denominator increase, but at different rates. The numerator \( n+2 \) grows linearly (like \( n \)), while the denominator \( n^2+1 \) grows quadratically (like \( n^2 \)).

2. **Divide by the Highest Power of \( n \)**:
   To make the behavior at infinity clearer, divide both the numerator and denominator by \( n^2 \) (the highest power of \( n \) in the denominator):

   \[
   \frac{n+2}{n^2+1} = \frac{\frac{n+2}{n^2}}{\frac{n^2+1}{n^2}} = \frac{\frac{n}{n^2} + \frac{2}{n^2}}{1 + \frac{1}{n^2}}.
   \]

3. **Simplify the Expression**:
   Simplifying each term, we get:

   \[
   = \frac{\frac{1}{n} + \frac{2}{n^2}}{1 + \frac{1}{n^2}}.
   \]

4. **Take the Limit as \( n \to \infty \)**:
   Now, take the limit as \( n \to \infty \):

   - \( \frac{1}{n} \to 0 \),
   - \( \frac{2}{n^2} \to 0 \),
   - \( \frac{1}{n^2} \to 0 \).

   So the expression becomes:

   \[
   \lim_{n \to \infty} \frac{0 + 0}{1 + 0} = \frac{0}{1} = 0.
   \]

### Conclusion:
Thus, we have shown that:

\[
\lim_{n \to \infty} \frac{n+2}{n^2 + 1} = 0.
\]

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Would you like more details on any of these steps, or do you have any further questions?

Here are 5 related questions to deepen your understanding:
1. How would the limit change if the numerator were \( n^2+2 \)?
2. What happens to the limit if the denominator were \( n+1 \) instead of \( n^2+1 \)?
3. Can we apply L'Hopital's Rule to this problem? Why or why not?
4. What is the limit of \( \frac{n^3 + 2}{n^4 + 1} \) as \( n \to \infty \)?
5. How would you compute the limit of a more complicated rational function?

**Tip**: Dividing both the numerator and denominator by the highest power of \( n \) helps simplify the behavior of limits at infinity.

Explore the detailed solution to finding the limit of the rational function (n + 2) / (n^2 + 1) as n approaches infinity. This step-by-step guide breaks down the limit behavior of rational functions using asymptotic analysis and simplification techniques. Perfect for advanced high school or introductory college calculus students.

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