Question

The value of $$ \lim_{x \to 0} (1 + 2x)^{1/x} $$ is

• A. 0
• B. \( e^2 \)
• C. \( e^{-2} \)
• D. \( e \)

Hint:

When you encounter a limit like \( (1 + u)^{1/u} \), recognize it as a form of the exponential limit that tends to \( e^2 \) if the expression inside the parentheses is \( 2x \).

Answer 1

The Correct Option is B

The limit in the question is of the form \( (1 + \text{something small})^{\text{something large}} \), which is a standard limit that approaches \( e^2 \) when simplified. To evaluate the limit, we use the following approximation: \[ \lim_{x \to 0} (1 + 2x)^{1/x} = e^{\lim_{x \to 0} \frac{\ln(1 + 2x)}{x}} \] Using the approximation \( \ln(1 + u) \approx u \) for small \( u \), we get: \[ \lim_{x \to 0} \frac{\ln(1 + 2x)}{x} = \lim_{x \to 0} \frac{2x}{x} = 2 \]
Thus, the limit is: \[ e^2 \]
Therefore, the correct answer is 2. \( e^2 \).