Question

Evaluate: \[ \lim_{n \to \infty} \sqrt{2} \left[ \frac{ \left(2 + \sqrt{2} \right)^n + \left(2 - \sqrt{2} \right)^n }{ \left(2 + \sqrt{2} \right)^n - \left(2 - \sqrt{2} \right)^n } \right] \]

• A. \( 2 + \sqrt{2} \)
• B. \( 2 - \sqrt{2} \)
• C. 1
• D. \( \sqrt{2} \)

Hint:

When expressions contain conjugate terms with exponential growth, the smaller decaying term becomes negligible for large \( n \).

Answer 1

The Correct Option is D

As \( n \to \infty \), \( \left(2 + \sqrt{2}\right)^n \gg \left(2 - \sqrt{2}\right)^n \) So, \[ \lim \approx \sqrt{2} \cdot \frac{(2 + \sqrt{2})^n}{(2 + \sqrt{2})^n} = \sqrt{2} \]