Question

Evaluate the following limit: $ \lim_{\theta \to 0} \frac{\theta \sin 2\theta}{1 - \cos 2\theta} $

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 4 \)

Hint:

For small angle limits, use the approximations \( \sin \theta \approx \theta \) and \( 1 - \cos \theta \approx \frac{\theta^2}{2} \).

Answer 1

The Correct Option is C

We are given: \[ \lim_{\theta \to 0} \frac{\theta \sin 2\theta}{1 - \cos 2\theta} \] We can use the small angle approximation \( \sin \theta \approx \theta \) and \( 1 - \cos \theta \approx \frac{\theta^2}{2} \) when \( \theta \) is small.
Thus, the limit becomes: \[ \lim_{\theta \to 0} \frac{\theta \cdot 2\theta}{\frac{2\theta^2}{2}} = \lim_{\theta \to 0} \frac{2\theta^2}{\theta^2} = 2 \]
Thus, the value of the limit is \( 2 \).