To find the limit of the function \( \frac{1}{n^2 + 1} \) as \( n \) approaches infinity, we analyze the behavior of the function as \( n \) becomes very large.

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

As \( n \) increases, \( n^2 \) grows much faster than 1. Thus, for very large \( n \), \( n^2 + 1 \) is approximately equal to \( n^2 \). This simplifies our expression:

\[
\frac{1}{n^2 + 1} \approx \frac{1}{n^2}
\]

Now, we can evaluate the limit of \( \frac{1}{n^2} \) as \( n \) approaches infinity:

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

Therefore, the limit of \( \frac{1}{n^2 + 1} \) as \( n \) approaches infinity is:

\[
\boxed{0}
\]

Would you like further details or have any questions about this explanation?

Here are some more questions you might be interested in:
1. What is the limit of \( \frac{1}{n^3 + 1} \) as \( n \) approaches infinity?
2. How do you find the limit of \( \frac{n}{n+1} \) as \( n \) approaches infinity?
3. What is the limit of \( \frac{2n + 3}{3n + 2} \) as \( n \) approaches infinity?
4. How do you evaluate the limit of \( \frac{1}{n} \) as \( n \) approaches infinity?
5. What is the limit of \( \frac{n^2 + 2n + 1}{n^2 + 1} \) as \( n \) approaches infinity?

**Tip:** When evaluating limits as \( n \) approaches infinity, compare the growth rates of the numerator and denominator to simplify the expression.

Discover how to find the limit of the function 1/(n^2 + 1) as n approaches infinity with a detailed step-by-step explanation. This problem involves analyzing limits, understanding asymptotic behavior, and applying the definition of limits. Suitable for advanced high school students.

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