The limit you're looking to evaluate is:

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

Step-by-step solution:

1. Recognize the limit form: The expression resembles the well-known limit definition of the exponential function. Specifically, we know that:

   $\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = e$

2. Rewrite the expression for easier understanding:

   We can express the given limit as:

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

   Since the limit $\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = e$, this becomes:

   $\lim_{n \to \infty} \left( e \right)^x = e^x$

Conclusion:

The limit is:

$e^x$

If you need more details or have any further questions, feel free to ask!

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Here are 5 related questions that can expand on this topic:

1. What happens if the expression is $\left( 1 + \frac{1}{n} \right)^{n x + 1}$?
2. How do you prove the limit $\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n = e$?
3. How does this formula relate to continuous compounding in finance?
4. What would happen if the base inside the parentheses were different, e.g., $\left( 1 + \frac{2}{n} \right)^n$?
5. How does this limit apply to approximating exponential growth processes?

Tip: When you encounter limits involving expressions like $\left( 1 + \frac{1}{n} \right)^n$, think of it as a way to approximate $e$ raised to different powers. This concept is fundamental in understanding continuous growth and compound interest.