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

This is a classic limit problem involving an exponential expression as \( x \) approaches infinity.

### Solution:

This limit resembles the well-known limit that defines the number \( e \):

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

In our case, instead of \( \frac{1}{x} \), we have \( \frac{2}{x} \). Thus, the limit becomes:

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

We can adjust the known result to account for the factor of 2. The result of this limit is \( e^2 \), since the extra factor of 2 causes the exponent to be multiplied by 2.

Thus, the solution is:

\[
e^2 \approx 7.389
\]

Would you like further details on the steps, or have any specific questions about this? Here are some related questions:

1. How is the number \( e \) derived in the context of limits?
2. Why does the factor of 2 in \( \frac{2}{x} \) change the result from \( e \) to \( e^2 \)?
3. What are some real-world applications of limits involving \( e \)?
4. How does this limit relate to compound interest formulas?
5. Can we apply similar limit techniques to other exponential functions?

**Tip:** When solving limit problems involving expressions of the form \( \left( 1 + \frac{k}{x} \right)^x \), always consider how it relates to the fundamental limit defining \( e \).

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

Explore how to solve the limit problem \( \lim_{x \to \infty} \left( 1 + \frac{2}{x} \right)^x \). This advanced calculus problem involves understanding exponential functions and the definition of Euler's number, suitable for Grades 11-12.

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