If
$$\lim_{x \rightarrow 0} \left[1 + x \ln (1 + b^2)\right]^{\frac{1}{x}} = 2b \sin \theta, b > 0 $$ and $$\theta \in (-\pi, \pi]$$, then the value of $$\theta$$ is

A

$$\pm \frac{\pi}{4}$$

B

$$\pm \frac{\pi}{3}$$

C

$$\pm \frac{\pi}{6}$$

D

$$\pm \frac{\pi}{2}$$