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We are asked to determine the limit of the following expression as \( n \to \infty \):

\[
\lim_{n \to \infty} \frac{n^3 + 1}{n^2 \sqrt{n^2 - 1}}.
\]

### Step-by-step solution:

1. **Simplify the expression:**

   Let's first simplify the square root term in the denominator:
   \[
   \sqrt{n^2 - 1} = n \sqrt{1 - \frac{1}{n^2}}.
   \]
   So, the original expression becomes:
   \[
   \lim_{n \to \infty} \frac{n^3 + 1}{n^2 \cdot n \sqrt{1 - \frac{1}{n^2}}} = \lim_{n \to \infty} \frac{n^3 + 1}{n^3 \sqrt{1 - \frac{1}{n^2}}}.
   \]
   
2. **Factor out \( n^3 \):**

   We now divide both terms in the numerator by \( n^3 \):
   \[
   \lim_{n \to \infty} \frac{n^3(1 + \frac{1}{n^3})}{n^3 \sqrt{1 - \frac{1}{n^2}}}.
   \]
   This simplifies to:
   \[
   \lim_{n \to \infty} \frac{1 + \frac{1}{n^3}}{\sqrt{1 - \frac{1}{n^2}}}.
   \]

3. **Evaluate the limit as \( n \to \infty \):**

   - As \( n \to \infty \), \( \frac{1}{n^3} \to 0 \), so \( 1 + \frac{1}{n^3} \to 1 \).
   - Similarly, \( \frac{1}{n^2} \to 0 \), so \( \sqrt{1 - \frac{1}{n^2}} \to \sqrt{1} = 1 \).
   
   Therefore, the expression simplifies to:
   \[
   \frac{1}{1} = 1.
   \]

### Final Answer:

\[
\lim_{n \to \infty} \frac{n^3 + 1}{n^2 \sqrt{n^2 - 1}} = 1.
\]

Would you like further details or explanations on any part of the solution?

Here are 5 related questions you might find useful:
1. How would the limit change if the denominator were \( n^2 \sqrt{n^2 + 1} \) instead of \( n^2 \sqrt{n^2 - 1} \)?
2. What is the behavior of the function when \( n \) approaches negative infinity?
3. How would you handle this limit using L'Hôpital's rule?
4. What are the steps to find the limit of an expression with more complex radicals in the denominator?
5. How does the convergence of this limit compare to other limits involving polynomials and radicals?

**Tip:** When dealing with limits involving radicals, it often helps to factor out the largest term inside the square root to simplify the expression.

Ermitteln Sie den Grenzwert: lim (n→∞) (n^3 + 1) / (n^2 * √(n^2 - 1))

Learn how to find the limit of (n^3 + 1) / (n^2 √(n^2 - 1)) as n approaches infinity. This problem covers limits, simplifying radicals, and applying limit laws. Suitable for students in Grades 10-12.

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