Step 1: Understanding the Concept:
This limit is the fundamental definition of the mathematical constant \( e \) (Euler's number). It represents a \( 1^\infty \) indeterminate form. Step 2: Key Formula or Approach:
For limits of the form \( \lim [f(x)]^{g(x)} \) where \( f(x) \to 1 \) and \( g(x) \to \infty \), the result is \( e^{\lim [f(x)-1]g(x)} \). Step 3: Detailed Explanation:
Let \( L = \lim_{x \to \infty} (1 + \frac{1}{x})^x \).
This is an indeterminate form of \( 1^\infty \). Using the formula:
\[ L = e^{\lim_{x \to \infty} [(1 + \frac{1}{x}) - 1] \cdot x} \]
\[ L = e^{\lim_{x \to \infty} (\frac{1}{x}) \cdot x} \]
\[ L = e^{\lim_{x \to \infty} 1} = e^1 = e \] Step 4: Final Answer:
The value of the limit is \( e \).