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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
$$\pm \frac{\pi}{4}$$
$$\pm \frac{\pi}{3}$$
$$\pm \frac{\pi}{6}$$
$$\pm \frac{\pi}{2}$$
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