Evaluate each limit. 


lim θ→0 (1/(2+sinθ)-1/2)/sin θ

Verified step by step guidance

1. Recognize that the given limit is in the indeterminate form 0/0 as \( \theta \to 0 \). This suggests the use of L'Hôpital's Rule, which is applicable for limits of the form 0/0 or \( \infty/\infty \).
2. Apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately. The original expression is \( \frac{\frac{1}{2+\sin\theta} - \frac{1}{2}}{\sin\theta} \).
3. Differentiate the numerator: The derivative of \( \frac{1}{2+\sin\theta} \) with respect to \( \theta \) is \( -\frac{\cos\theta}{(2+\sin\theta)^2} \). The derivative of \( \frac{1}{2} \) is 0.
4. Differentiate the denominator: The derivative of \( \sin\theta \) with respect to \( \theta \) is \( \cos\theta \).
5. Substitute the derivatives back into the limit expression: \( \lim_{\theta \to 0} \frac{-\frac{\cos\theta}{(2+\sin\theta)^2}}{\cos\theta} \). Simplify the expression and evaluate the limit as \( \theta \to 0 \).