Question

$\lim_{n\to\infty} (1 + \frac{1}{n})^{2n}$ is equal to}

• A. e
• B. $e^2$
• C. $e^3$
• D. 4.0

Hint:

Remember the general rule for limits of this type:
\[ \lim_{n\to\infty} \left(1 + \frac{a}{n}\right)^{bn} = e^{ab} \] Applying this rule to our problem where \(a = 1\) and \(b = 2\):
\[ \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^{2n} = e^{1 \times 2} = e^2 \] This general formula allows you to solve similar limits instantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We evaluate the limit of an indeterminate form of the type \(1^\infty\) as \(n \to \infty\).
The mathematical constant \(e\) (Euler's number) is defined using this limit.

Step 2: Key Formula or Approach:
The standard limit definition for the constant \(e\) is:
\[ \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n = e \] Using the power laws for limits, if \(\lim x_n = L\), then:
\[ \lim (x_n)^k = \left(\lim x_n\right)^k \]

Step 3: Detailed Explanation:
Let us calculate the limit step-by-step:
1. We are asked to evaluate the limit:
\[ L = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^{2n} \] 2. Using exponent rules, we can rewrite the expression as:
\[ \left(1 + \frac{1}{n}\right)^{2n} = \left[ \left(1 + \frac{1}{n}\right)^n \right]^2 \] 3. Applying limit laws:
\[ L = \left[ \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n \right]^2 \] 4. Substituting the standard limit definition:
\[ L = [e]^2 = e^2 \] Therefore, the limit is equal to \(e^2\).

Step 4: Final Answer:
The value of the limit is \(e^2\), matching Option (B).