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  4. displaystyle lim x arrow (π/2)( tan 2 x((2 sin 2 x+3 sin x+4)(1/2)-( sin 2 x+6 sin x+2)(1/2))) is equal to

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Q. $\displaystyle\lim _{x \rightarrow \frac{\pi}{2}}\left(\tan ^{2} x\left(\left(2 \sin ^{2} x+3 \sin x+4\right)^{\frac{1}{2}}-\left(\sin ^{2} x+6 \sin x+2\right)^{\frac{1}{2}}\right)\right)$ is equal to

JEE MainJEE Main 2022Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \tan ^{2} x\left[\sqrt{2 \sin ^{2} x+3 \sin x+4}-\sqrt{\sin ^{2} x+6 \sin x+2}\right]=$
$\displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan ^{2} x\left[\sin ^{2} x-3 \sin x+2\right]}{\sqrt{9}+\sqrt{9}}$
$=\displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan ^{2} x(\sin x-1)(\sin x-2)}{6}$
$=\frac{1}{6} \displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \tan ^{2} x(1-\sin x)$
$=\frac{1}{6} \displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin ^{2} x(1-\sin x)(1+\sin x)}{(1-\sin x)}=\frac{1}{12}$