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If $lim_{x \to 0} {(1 + x \log (1 + b^{2}))}^{1 / x} = 2 b \sin^{2} θ$ , $b > 0 and θ \in [- π, π]$ , then the value of $θ$ is

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Correct option is C

limx→0 1+xlog⁡1+b21/x=limx→0 1+xlog⁡1+b21xlog⁡1+b2×log⁡1+b2=elog⁡1+b2=1+b2Thus ,1+b2=2bsin2⁡θ⇒ sin2⁡θ=1+b22b≥1⇒ sin2⁡θ=1⇒sin⁡θ=±1 or θ=±π/2

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If limx→0 1+xlog⁡1+b21/x=2bsin2⁡θ ,b>0 and θ∈[−π,π], then the value of θ is

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If $lim_{x \to 0} {(1 + x \log (1 + b^{2}))}^{1 / x} = 2 b \sin^{2} θ$ , $b > 0 and θ \in [- π, π]$ , then the value of $θ$ is