Step 1: Understanding the Question:
This question involves evaluating a fundamental limit in calculus that arises from the study of continuous compounding and growth.
The expression is of the form \( 1^\infty \), which is an indeterminate form.
This particular limit is used to define Euler's number, denoted as \( e \).
Step 2: Key Formula or Approach:
1. If a limit is of the form \( \lim_{x \to a} [f(x)]^{g(x)} \) where \( f(x) \to 1 \) and \( g(x) \to \infty \), it can be evaluated as \( e^{\lim_{x \to a} (f(x) - 1) g(x)} \).
2. Alternatively, recognize this as the standard definition of the exponential constant \( e \).
Step 3: Detailed Explanation:
Let the limit be \( L = \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x \).
As \( x \) approaches infinity, \( 1/x \) approaches 0. Thus the base \( (1 + 1/x) \) approaches 1.
The exponent \( x \) approaches infinity. This gives us the \( 1^\infty \) form.
Let's use the natural logarithm to simplify:
\[ \ln L = \lim_{x \to \infty} x \ln\left(1 + \frac{1}{x}\right) \]
Rewrite as:
\[ \ln L = \lim_{x \to \infty} \frac{\ln(1 + 1/x)}{1/x} \]
Let \( t = 1/x \). As \( x \to \infty \), \( t \to 0 \).
\[ \ln L = \lim_{t \to 0} \frac{\ln(1 + t)}{t} \]
Using the standard limit property \( \lim_{t \to 0} \frac{\ln(1 + t)}{t} = 1 \):
\[ \ln L = 1 \]
Taking the exponential of both sides:
\[ L = e^1 = e \]
Therefore, the limiting value of the growth sequence is the irrational constant \( e \approx 2.71828 \).
Step 4: Final Answer:
The value of the limit is \( e \).