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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 $e$.