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The given expression is:

\[
\lim_{n \to \infty} \left( \frac{n + 1}{n - 1} \right)^n
\]

To solve this, we can simplify the expression inside the limit as \( n \to \infty \).

### Step-by-step solution:

1. **Rewrite the expression**:
   \[
   \frac{n+1}{n-1} = \frac{n(1 + \frac{1}{n})}{n(1 - \frac{1}{n})} = \frac{1 + \frac{1}{n}}{1 - \frac{1}{n}}
   \]

2. **Use the limit for large \(n\)**:
   As \( n \to \infty \), the terms \( \frac{1}{n} \to 0 \). Therefore, the expression simplifies to:
   \[
   \frac{1 + \frac{1}{n}}{1 - \frac{1}{n}} \approx \frac{1 + 0}{1 - 0} = 1
   \]
   However, to accurately compute the limit of the entire expression, we need a more precise expansion around \( n \to \infty \).

3. **Expand using a binomial approximation**:
   For large \(n\), we use the binomial approximation to express:
   \[
   \frac{1 + \frac{1}{n}}{1 - \frac{1}{n}} \approx 1 + \frac{2}{n}
   \]

4. **Exponentiate**:
   Now we return to the original limit:
   \[
   \left(1 + \frac{2}{n}\right)^n
   \]
   As \( n \to \infty \), this expression is of the form \( \left(1 + \frac{a}{n}\right)^n \), which tends to \( e^a \) by the definition of the exponential function. Here, \( a = 2 \).

5. **Conclusion**:
   Thus, the limit becomes:
   \[
   \lim_{n \to \infty} \left(1 + \frac{2}{n}\right)^n = e^2
   \]

### Final Answer:
\[
\lim_{n \to \infty} \left( \frac{n + 1}{n - 1} \right)^n = e^2
\]

Would you like more details or have any questions?

Here are five follow-up questions to deepen your understanding:
1. How does the binomial approximation help simplify limits involving large \(n\)?
2. Can you explain why \( \left(1 + \frac{a}{n}\right)^n \) tends to \(e^a\) as \( n \to \infty \)?
3. What other types of limits result in expressions involving the exponential function \(e\)?
4. How would you approach the limit if the exponent on the fraction were not \(n\) but a different variable?
5. What are other common approximations for limits involving rational functions?

**Tip:** When dealing with limits at infinity, it’s helpful to explore expansions or approximations, like the binomial theorem, to simplify complex expressions.

Explore the limit of the expression ((n+1)/(n-1))^n as n approaches infinity. This problem involves exponential growth, binomial approximation, and the limit theorem at infinity, making it suitable for Grades 10-12 students.

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