Question

What is the value of the limit \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \)?

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

Hint:

Always remember: \[ \left(1 + \frac{1}{n}\right)^n \rightarrow e \] A standard and very important limit in calculus.

Answer 1

The Correct Option is C

Concept: Definition of \(e\)
The number \(e\) is defined as: \[ e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \]
Step 1: Recognize the standard limit
The given expression is a direct standard limit used to define \(e\).
Step 2: Intuition behind the limit
This expression arises in:

• Compound interest problems
• Continuous growth models

As \(n\) increases, the quantity: \[ \left(1 + \frac{1}{n}\right)^n \] approaches a fixed number.
Step 3: Numerical idea
\[ n=1 \Rightarrow 2,\quad n=2 \Rightarrow 2.25,\quad n=10 \Rightarrow 2.593,\quad n \to \infty \Rightarrow 2.718... \]
Step 4: Final value
\[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.718 \] Conclusion: \[ \text{Limit value is } e \]