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The given mathematical expression is:

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

Let's simplify and evaluate this limit step-by-step.

### Step 1: Rewrite the expression
The expression inside the limit can be rewritten as:

\[
\frac{1 + x}{x} = 1 + \frac{1}{x}
\]
Therefore, the original limit becomes:
\[
\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{2x}
\]

### Step 2: Recognize the limit form
We know that as \( x \to \infty \), the expression \( \left(1 + \frac{1}{x}\right)^x \to e \). Thus, 
\[
\left(1 + \frac{1}{x}\right)^{2x} = \left[\left(1 + \frac{1}{x}\right)^x\right]^2
\]

As \( x \to \infty \), \( \left(1 + \frac{1}{x}\right)^x \to e \), hence

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

### Conclusion
The limit is:
\[
\lim_{x \to \infty} \left(\frac{1 + x}{x}\right)^{2x} = e^2
\]

The final answer is \( e^2 \).

Would you like further details or explanations?

### Related Questions
1. What is the limit of \( \left(1 + \frac{1}{n}\right)^n \) as \( n \to \infty \)?
2. How would the limit change if the power was \( 3x \) instead of \( 2x \)?
3. What is the derivative of \( \left(\frac{1 + x}{x}\right)^{2x} \) with respect to \( x \)?
4. How does this problem relate to the definition of the number \( e \)?
5. How would the limit change if \( x \to 0 \) instead of \( x \to \infty \)?

### Tip
The expression \( \left(1 + \frac{1}{x}\right)^x \) is a fundamental representation of the number \( e \). Understanding this helps in grasping more complex exponential limits!

Find the limit of the expression [(1 + x) / x]^(2x) as x approaches infinity using key limit and exponential concepts. This problem illustrates the use of the limit definition of e and is suitable for students in Grades 11-12 or early college-level calculus.

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