Step 1: Understanding the Question
The question asks for the value of the limit of the expression \( \left(1 + \frac{1}{n}\right)^n \) as the variable \(n\) approaches infinity.
Step 2: Key Formula or Approach
This limit is one of the fundamental definitions 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
The expression is a standard form in calculus. As \(n\) becomes very large, the term \( \frac{1}{n} \) approaches 0. This creates an indeterminate form of the type \(1^\infty\). The resolution of this specific indeterminate form is the definition of \(e\). The value of \(e\) is an irrational number approximately equal to 2.71828.
For competitive exams, this is a standard result that should be memorized.
Step 4: Final Answer
The limit of \( \left(1 + \frac{1}{n}\right)^n \) as \(n \to \infty\) is \(e\).