Question

Find the limit of \( \left(1 + \frac{1}{n}\right)^n \) as \(n \to \infty\).

• A. \(1\)
• B. \(e\)
• C. \(0\)
• D. \(2\)

Hint:

A very important limit in calculus is: \[ \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e \] This limit is used in compound interest, exponential growth, and many areas of mathematics.

Answer 1

The Correct Option is B

Concept:
The limit \[ \lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n \] is a well-known fundamental limit in calculus. It defines the mathematical constant \(e\), which is the base of natural logarithms. \[ e \approx 2.71828 \] Thus, \[ \lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e \]
Step 1: Identify the standard limit form.
The expression given in the question exactly matches the standard definition of \(e\): \[ e = \lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n \]
Step 2: Apply the known result.
Since the expression is the same as the definition of \(e\), \[ \lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e \]
Step 3: State the final result.
\[ \therefore \text{The value of the limit is } e. \]