Question

If \( \lim_{x \to \infty} \left( 1 + \frac{a}{x} + \frac{b}{x^2} \right)^{2x} = e^2 \), then

• A. \( a = 1, b = 2 \)
• B. \( a = 2, b = 1 \)
• C. \( a = 1, b \in \mathbb{R} \)
• D. None of these

Hint:

When dealing with limits of the form \( \left( 1 + \frac{1}{n} \right)^n \), use the approximation \( \left( 1 + \frac{a}{x} \right)^x \approx e^a \) for large \( x \).

Answer 1

The Correct Option is C