Question

Find the limit \[ \lim_{x \to 0} \left( 1 + \tan^2 \sqrt{x} \right)^{3/x} \]

• A. 0
• B. \( \infty \)
• C. \( e \)
• D. \( e^3 \)

Hint:

In limits involving small angles and powers, simplify using standard approximations such as \( \tan(\theta) \approx \theta \) for small \( \theta \).

Answer 1

The Correct Option is D

Using the approximation \( \tan^2(\sqrt{x}) \approx x \) for small \( x \), the limit simplifies to: \[ \lim_{x \to 0} \left( 1 + x \right)^{3/x} = e^3 \]