To determine the point of convergence (limit) of the sequence $X_n = \frac{n^2 + 1}{n^4 - 2}$, we need to analyze its behavior as $n \to \infty$.

Step 1: Look at the dominant terms

As $n$ becomes very large, the highest degree terms in the numerator and denominator will dominate the behavior of the sequence. So, we focus on the leading terms:

• Numerator: $n^2 + 1$, where $n^2$ is the dominant term.
• Denominator: $n^4 - 2$, where $n^4$ is the dominant term.

Thus, for large $n$, we can approximate the sequence as: $X_n \approx \frac{n^2}{n^4}$ This simplifies to: $X_n \approx \frac{1}{n^2}$

Step 2: Take the limit

Now, we evaluate the limit as $n \to \infty$: $\lim_{n \to \infty} \frac{1}{n^2} = 0$

Conclusion

The sequence converges to 0 as $n \to \infty$. Therefore, the point of convergence (limit) is: $\boxed{0}$

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Related Questions:

1. How do you apply L'Hopital's Rule to sequences like this?
2. What happens if the numerator has a higher degree than the denominator in a sequence?
3. How can you determine the limit of a sequence with an exponential term?
4. What are the steps to find the limit of a sequence involving trigonometric functions?
5. How does the behavior of rational functions change with respect to infinity?

Tip: When analyzing the limit of rational functions as $n \to \infty$, focus on the highest degree terms in the numerator and denominator for simplification.