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  4. The value of displaystyle lim x arrow 0((x/√[8]1- sin x-√[8]1+ sin x)) is equal to:

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Q. The value of $\displaystyle\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to:

JEE MainJEE Main 2021Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$
$=\displaystyle\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$
$=\displaystyle\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$
$\left(\frac{(\sqrt[8]{1-\sin x}+\sqrt[8]{1+\sin x})}{\sqrt[8]{1-\sin x}+\sqrt[8]{1+\sin x}}\right)\left(\frac{(\sqrt[4]{1-\sin x}+\sqrt[4]{1+\sin x})}{\sqrt[4]{1-\sin x}+\sqrt[4]{1+\sin x}}\right)$
$\left(\frac{(\sqrt[2]{1-\sin x}+\sqrt[2]{1+\sin x})}{\sqrt[2]{1-\sin x}+{ }^{2} \sqrt{1+\sin x}}\right)=\displaystyle\lim _{x \rightarrow 0}\left(\frac{x}{1-\sin x-(1+\sin x)}\right)$
$(\sqrt[8]{1-\sin x}+\sqrt[8]{1+\sin x})(\sqrt[4]{1-\sin x}+\sqrt[4]{1+\sin x})\left(\sqrt[2]{1-\sin x}+{ }^{2} \sqrt{1+\sin x}\right)$
$=\displaystyle\lim _{x \rightarrow 0}(\sqrt[8]{1-\sin x}+\sqrt[8]{1+\sin x})(\sqrt[4]{1-\sin x}+\sqrt[4]{1+\sin x})$
$(\sqrt[2]{1-\sin x}+\sqrt[2]{1+\sin x})$
$=\displaystyle\lim _{x \rightarrow 0}\left(-\frac{1}{2}\right)(2)(2)(2)\left\{\because \displaystyle\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right\}=-4$