We are tasked with evaluating the limit of the following expression as \( n \) approaches infinity:

\[
\lim_{n \to \infty} \left( 1 - \frac{1}{n} \right)^{2n}
\]

### Solution:

We recognize that this limit is closely related to the famous limit definition of the number \( e \), which states:

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

Thus, the expression we are given involves a small modification to this standard limit. Specifically, it involves \( 2n \) instead of \( n \), so we can use the following relationship:

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

### Final Answer:

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

Let me know if you'd like any further details or have any other questions!

Here are 5 related questions to expand on this concept:

1. What is the derivation of the limit definition of \( e \)?
2. How does the binomial theorem help approximate limits of the form \( \left(1 + \frac{x}{n}\right)^n \)?
3. Can you prove the limit \( \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^{n} = e^x \)?
4. How does L'Hôpital's Rule apply to limits involving indeterminate forms like \( \frac{0}{0} \)?
5. What are the implications of this limit in probability theory (e.g., binomial distribution approximation)?

**Tip**: Limits involving exponential expressions often arise in applications of continuous growth models, especially in economics and biology.

Explore the solution for the limit lim n→∞ (1 - 1/n)²ⁿ, where we apply the limit definition of e and show that the result converges to 1/e². This calculus problem is ideal for students familiar with limits and exponential functions.

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