$\lim _{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos x}}{\tan ^2 \frac{x}{2}}=$

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos x}}{\tan ^2 \frac{x}{2}}=$
  1. 1
  2. 2
  3. 3
  4. $-1$

Solution

$\begin{aligned} & \lim _{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos x}}{\tan ^2 \frac{x}{2}} \\ & =\lim _{x \rightarrow 0} \frac{1+x \sin x-\cos x}{\tan ^2 \frac{x}{2}(\sqrt{1+x \sin x}+\sqrt{\cos x})}\end{aligned}$ $\begin{aligned} & =\lim _{x \rightarrow 0} \frac{x \sin x+2 \sin ^2 \frac{x}{2}}{\tan ^2 \frac{x}{2}(\sqrt{1+x \sin x}+\sqrt{\cos x})} \\ & =\lim _{x \rightarrow 0} \frac{\frac{\sin x}{x}+\frac{2 \sin ^2 \frac{x}{2}}{x^2}}{\frac{\tan ^2 \frac{x}{2}}{x^2}(\sqrt{1+x \sin x}+\sqrt{\cos x})}\end{aligned}$ $=\lim _{x \rightarrow 0}\left(\frac{\frac{\sin x}{x}+\frac{2 \sin ^2 \frac{x}{2}}{\frac{x^2}{4}} \times \frac{1}{4}}{\frac{\tan ^2 \frac{x}{2}}{\frac{x^2}{4}} \times \frac{1}{4}(\sqrt{1+x \sin x}+\sqrt{\cos x})}\right)$ $\begin{aligned} & =\frac{1+2 \times 1 \times \frac{1}{4}}{1 \times \frac{1}{4}(\sqrt{1+0}+\sqrt{1})} \\ & =3\end{aligned}$

Asked in: MHT CET 2023 (13 May Shift 2)

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