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Question Number 83361 by john santu last updated on 01/Mar/20

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

Commented by john santu last updated on 01/Mar/20

$$\underset{x\to 0^{+}}{\lim }\frac{1-(1-\frac{1}{2}{\mathrm{x}}^2+\mathrm{o}\left({\mathrm{x}}^2\right)).\sqrt{\mathrm{x}}}{\sqrt{\mathrm{x}}.\sqrt{\mathrm{x}}}$$$$=\underset{x\to 0^{+}}{\lim }\sqrt{\mathrm{x}}\times \underset{x\to 0^{+}}{\lim }\frac{\frac{1}{2}{\mathrm{x}}^2-\mathrm{o}\left({\mathrm{x}}^2\right)}{{\mathrm{x}}^2}$$$$=0\times \frac{1}{2}=0$$

Answered by mr W last updated on 01/Mar/20

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

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