https://r2.erweima.ai/mathbot/image/17249267204679tx0iqql.jpg

The image presents the limit:

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

To prove this limit, we'll proceed with the following steps:

### Step 1: Understand the Expression
The expression inside the limit is \(\frac{1}{{n^2 + 1}}\). As \(n\) becomes very large (i.e., as \(n \to \infty\)), the term \(n^2 + 1\) also becomes very large because \(n^2\) dominates over the constant 1.

### Step 2: Behavior as \(n\) Approaches Infinity
Since \(n^2 + 1\) grows without bound as \(n\) increases, the fraction \(\frac{1}{{n^2 + 1}}\) becomes smaller and smaller. Specifically:

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

As \(n \to \infty\), \(\frac{1}{{n^2}}\) approaches 0 because the denominator increases without bound.

### Step 3: Formal Proof
For any \(\epsilon > 0\), we need to find an \(N > 0\) such that for all \(n > N\):

\[
\left|\frac{1}{{n^2 + 1}} - 0\right| < \epsilon
\]

This simplifies to:

\[
\frac{1}{{n^2 + 1}} < \epsilon
\]

Since \(n^2 + 1 > n^2\), we have:

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

So, we require:

\[
\frac{1}{{n^2}} < \epsilon
\]

This implies:

\[
n^2 > \frac{1}{\epsilon}
\]

Taking the square root of both sides:

\[
n > \frac{1}{\sqrt{\epsilon}}
\]

Thus, for any \(\epsilon > 0\), if we choose \(N = \frac{1}{\sqrt{\epsilon}}\), then for all \(n > N\), \(\frac{1}{{n^2 + 1}} < \epsilon\).

### Conclusion
Therefore:

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

This completes the proof.

Would you like any additional details or have any further questions?

Here are five related questions you might find useful:

1. What is the limit of \(\frac{1}{n^2}\) as \(n\) approaches infinity?
2. How do limits help determine the convergence of sequences?
3. What are some other examples of limits that approach zero as \(n\) approaches infinity?
4. How do you apply the \(\epsilon-N\) definition of a limit in more complex problems?
5. How would the limit change if the expression were \(\frac{1}{n^2 - 1}\)?

**Tip:** Understanding the behavior of functions as they approach infinity is crucial for grasping concepts in calculus, especially when working with limits, series, and integrals.

Learn how to prove the limit of 1/(n^2 + 1) as n approaches infinity is 0 with a detailed step-by-step proof. This problem covers key calculus concepts, including limits and the epsilon-delta definition. Suitable for advanced high school and college students.

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