Question

If $\displaystyle\lim _{x \rightarrow 0}\left[1+x \ln \left(1+b^{2}\right)\right]^{1 / x}=2 b \sin ^{2} \theta, b>0$ 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. $\pm \frac{\pi}{2}$

Answer 1

Answer: (D)

$e ^{\ln \left(1+ b ^{2}\right)}=2 b \sin ^{2} \theta$
$\Rightarrow \sin ^{2} \theta=\frac{1+ b ^{2}}{2 b }$
$\Rightarrow \sin ^{2} \theta=1$ as $\frac{1+ b ^{2}}{2 b } \geq 1$
$\theta=\pm \pi / 2$