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Question Number 84492 by john santu last updated on 13/Mar/20

$$\underset{x\to \frac{\pi }{3}}{\lim }\frac{\sin (x-\frac{\pi }{3})}{1-2\cos \left(x\right)}=$$

Commented by john santu last updated on 13/Mar/20

$$\underset{x\to 0}{\lim }\frac{\sin \left(x\right)}{1-2\cos (x+\frac{\pi }{3})}=$$$$\underset{x\to 0}{\lim }\frac{\cos \left(x\right)}{2\sin (x+\frac{\pi }{3})}=\frac{1}{2\times \frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}$$

Commented by john santu last updated on 13/Mar/20

$$\mathrm{other}\mathrm{way}$$$$\sin (\mathrm{x}-\frac{\pi }{3})\approx (\mathrm{x}-\frac{\pi }{3})$$$$\cos \left(\mathrm{x}\right)\approx \frac{1}{2}-\frac{\sqrt{3}}{2}(\mathrm{x}-\frac{\pi }{3})$$$$\underset{x\to \frac{\pi }{3}}{\lim }\frac{x-\frac{\pi }{3}}{1-2[\frac{1}{2}-\frac{\sqrt{3}}{2}(\mathrm{x}-\frac{\pi }{3})]}=$$$$\underset{x\to \frac{\pi }{3}}{\lim }\frac{(\mathrm{x}-\frac{\pi }{3})}{\sqrt{3}(\mathrm{x}-\frac{\pi }{3})}=\frac{1}{\sqrt{3}}.$$

Commented by mathmax by abdo last updated on 13/Mar/20

$$letl\left(x\right)=\frac{sin(x-\frac{\pi }{3})}{1-2cosx}changementx-\frac{\pi }{3}=tgive$$$$l\left(x\right)=A\left(t\right)=\frac{sint}{1-2cos(t+\frac{\pi }{3})}=\frac{sint}{1-2(costcos\frac{\pi }{3}-sintsin\left(\frac{\pi }{3}\right))}$$$$=\frac{sint}{1-2(\frac{1}{2}cost-\frac{\sqrt{3}}{2}sint)}=\frac{sint}{1-cost+\sqrt{3}sint}$$$$x\to \frac{\pi }{3}\iff t\to 0andA\left(t\right)\sim \frac{t}{\frac{t^2}{2}+\sqrt{3}t}=\frac{1}{\frac{t}{2}+\sqrt{3}}\Rightarrow {lim}_{t\to 0}A\left(t\right)=\frac{1}{\sqrt{3}}$$$$\Rightarrow {lim}_{x\to 0}l\left(x\right)=\frac{1}{\sqrt{3}}$$

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