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Question Number 192651 by beto last updated on 24/May/23

$$$$$${lim}_{x\to 0}\frac{2-\sqrt{cos\left(x\right)}-cos\left(x\right)}{x^2}$$

Answered by cortano12 last updated on 24/May/23

$$\mathrm{L}=\underset{x\to 0}{\lim }\frac{(1-\sqrt{\cos \mathrm{x}})+(1-\cos \mathrm{x})}{{\mathrm{x}}^2}$$$$\mathrm{L}=\underset{x\to 0}{\lim }\frac{1-\cos \mathrm{x}}{{2\mathrm{x}}^2}+\underset{x\to 0}{\lim }\frac{1-\cos \mathrm{x}}{{\mathrm{x}}^2}$$$$\mathrm{L}=\underset{x\to 0}{\lim }\frac{3(1-\cos \mathrm{x})}{{2\mathrm{x}}^2}$$$$\mathrm{L}=\underset{x\to 0}{\lim }\frac{3.\sin {}^2\mathrm{x}}{{4\mathrm{x}}^2}=\frac{3}{4}$$

Answered by Subhi last updated on 24/May/23

$${\lim }_{\mathrm{x}\to 0}\frac{1}{2}+\frac{1-\sqrt{cos\left(x\right)}}{x^2}.\frac{1+\sqrt{cos\left(x\right)}}{1+\sqrt{cos\left(x\right)}}$$$$\frac{1}{2}+\frac{1-cos\left(x\right)}{x^2.(1+\sqrt{cos\left(x\right)})}$$$$\frac{1}{2}+\frac{2{sin}^2\left(\frac{x}{2}\right)}{x^2(1+\sqrt{cos\left(x\right)})}$$$${lim}_{x\to 0}(\frac{1}{2}+\frac{1}{2.(1+\sqrt{cos\left(x\right)})})=\frac{1}{2}+\frac{1}{4}=\frac{3}{4}$$

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