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Question Number 143960 by bobhans last updated on 20/Jun/21

$$\mathrm{The}\mathrm{value}\mathrm{of}\underset{x\to 0}{\lim }\frac{\sqrt{1-\cos {\mathrm{x}}^2}}{1-\cos \mathrm{x}}=?$$

Answered by lapache last updated on 20/Jun/21

$$li\underset{x\to 0}{m}\frac{\sqrt{1-1+\frac{x^4}{2}}}{1-1+\frac{x^2}{2}}=li\underset{x\to 0}{m}\frac{2\sqrt{\frac{x^4}{2}}}{x^2}=\sqrt{2}$$

Answered by bramlexs22 last updated on 20/Jun/21

$$\underset{x\to 0}{\lim }\frac{\sqrt{2\sin {}^2\left(\frac{x^2}{2}\right)}}{2\sin {}^2\left(\frac{x}{2}\right)}=\frac{\sqrt{2}}{2}\underset{x\to 0}{\lim }\frac{\sin \left(\frac{x^2}{2}\right)}{\sin {}^2\left(\frac{x}{2}\right)}$$$$=\frac{2\sqrt{2}}{2}.1=\sqrt{2}$$

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