This question asks for the value of one of the most famous and fundamental limits in calculus. This limit is used as a formal definition for a very important mathematical constant.
Step 1: Understanding the Question:
We are asked to evaluate the limit of the sequence \( a_n = (1 + \frac{1}{n})^n \) as \( n \) approaches infinity.
Step 2: Key Formula or Approach:
The expression is the standard limit definition of the mathematical constant \(e\), the base of the natural logarithm.
\[ e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \]
Step 3: Detailed Explanation:
This limit represents an indeterminate form of type \(1^\infty\). As \(n \to \infty\), the base \( (1 + \frac{1}{n}) \) approaches 1, while the exponent \(n\) approaches infinity. The value of the limit is the result of the balance between the base getting closer to 1 and the exponent getting larger.
This limit is formally taken as the definition of the number \(e\). It arises naturally in contexts of continuous growth, such as continuously compounded interest. The value of \(e\) is an irrational number approximately equal to 2.71828.
Step 4: Final Answer:
The value of the limit is \(e\).