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Question Number 116417 by bemath last updated on 03/Oct/20

$$\underset{x\to 0}{\lim }\frac{\sqrt{1+\mathrm{x}\sin \mathrm{x}}-\sqrt{\cos 2\mathrm{x}}}{\tan {}^2\left(\frac{\mathrm{x}}{2}\right)}=?$$

Answered by Bird last updated on 03/Oct/20

$$f\left(x\right)=\frac{\sqrt{1+xsinx}-\sqrt{cos\left(2x\right)}}{{tan}^2\left(\frac{x}{2}\right)}$$$$wehsve\sqrt{1+xsinx}\sim \sqrt{1+x^2}\sim 1+\frac{x^2}{2}$$$$\sqrt{cos\left(2x\right)}\sim \sqrt{1-2x^2}\sim 1-x^2$$$${tan}^2\left(\frac{x}{2}\right)\sim \frac{x^2}{4}\Rightarrow$$$$f\left(x\right)\sim \frac{1+\frac{x^2}{2}-1+x^2}{\frac{x^2}{4}}\Rightarrow$$$$f\left(x\right)\sim 4.\frac{\frac{3}{2}{x}^2}{x^2}=6\Rightarrow$$$${lim}_{x\to 0}f\left(x\right)=6$$$$$$

Answered by bobhans last updated on 04/Oct/20

$$\underset{x\to 0}{\lim }\frac{(1+\mathrm{xsin}\mathrm{x})-(1-2\sin {}^2\mathrm{x})}{\tan {}^2\left(\frac{1}{2}\mathrm{x}\right)}\times \underset{x\to 0}{\lim }\frac{1}{\sqrt{1+\mathrm{xsin}\mathrm{x}}+\sqrt{\cos 2\mathrm{x}}}=$$$$\underset{x\to 0}{\lim }\frac{\mathrm{xsin}\mathrm{x}+2\sin {}^2\mathrm{x}}{\tan {}^2\left(\frac{1}{2}\mathrm{x}\right)}\times \frac{1}{2}=$$$$\frac{1}{2}\times \underset{x\to 0}{\lim }\frac{[\frac{\mathrm{x}\sin \mathrm{x}}{{\mathrm{x}}^2}+\frac{2\sin {}^2\mathrm{x}}{{\mathrm{x}}^2}]}{\left[\frac{\tan {}^2\left(\frac{1}{2}\mathrm{x}\right)}{{\mathrm{x}}^2}\right]}=\frac{1}{2}\times \frac{3}{\left(\frac{1}{4}\right)}=6.$$

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