The given problem is to evaluate the limit:

$\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x$

This is a classic problem involving exponential limits.

Step-by-Step Solution:

1. Recognize the structure of the limit: The expression $\left(1 + \frac{2}{x}\right)^x$ resembles the form used in the definition of the exponential function. Specifically: $\left(1 + \frac{k}{x}\right)^x \to e^k \quad \text{as } x \to \infty.$

2. Compare to the general formula: In this case, $k = 2$. Therefore: $\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x = e^2.$

3. Simplify the final answer: The limit evaluates to: $e^2.$

Final Answer:

$\boxed{e^2}$

Would you like a detailed derivation or explanation of why this formula works?

Here are 5 related questions to expand your understanding:

1. How does the limit $\lim_{x \to \infty} \left(1 + \frac{k}{x}\right)^x = e^k$ originate from the definition of $e$?
2. What happens if the power of $x$ in the problem changes to $x^2$? How does that affect the solution?
3. Can this technique be applied to expressions where the numerator is not constant, such as $\frac{2x}{x}$?
4. How does this problem relate to the compound interest formula in finance?
5. How would you compute the numerical value of $e^2$ to 3 decimal places?

Tip: The expression $\left(1 + \frac{k}{x}\right)^x$ is a foundational example of the exponential function in calculus. Understanding its derivation and applications can help in solving many similar problems.