If lim x 0 [ 1 + x ln ( 1 + b 2 ) ] 1 / x = 2 b sin 2 , b 0 and , then the value of is [2011]

Question

If $\lim_{x \to 0} {[1 + x \ln (1 + b^{2})]}^{1 / x} = 2 b \sin^{2} θ, b > 0$ and $θ \in (- π, π],$ then the value of $θ$ is [2011]

Smart Achievers · Accepted Answer

(4)

$\lim_{x \to 0} {[1 + x \ln (1 + b^{2})]}^{\frac{1}{x}} = 2 b \sin^{2} θ$

$\Rightarrow e^{\lim_{x \to 0} \frac{1}{x} \ln [1 + x \ln (1 + b^{2})]} = 2 b \sin^{2} θ$

$\Rightarrow e^{\lim_{x \to 0} \frac{\ln [1 + x \ln (1 + b^{2})]}{x \ln (1 + b^{2})} \times \ln (1 + b^{2})} = 2 b \sin^{2} θ$

$\Rightarrow e^{\ln (1 + b^{2})} = 2 b \sin^{2} θ$

$\Rightarrow 1 + b^{2} = 2 b \sin^{2} θ$ $\Rightarrow 2 \sin^{2} θ = b + \frac{1}{b}$

We know that $2 \sin^{2} θ \leq 2$ and $b + \frac{1}{b} \geq 2 for b > 0$

$∴$ $2 \sin^{2} θ = b + \frac{1}{b} = 2 \Rightarrow \sin^{2} θ = 1$

$∵$ $θ \in (- π, π]$ $∴$ $θ = \pm \frac{π}{2}$

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