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.