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Question Number 119961 by bobhans last updated on 28/Oct/20

$$Without{L}^{\prime }Hopitalrule$$$$\underset{x\to \pi /3}{\lim }\frac{\sin (x-\frac{\pi }{3})}{1-2\cos x}?$$

Answered by bramlexs22 last updated on 28/Oct/20

Answered by Dwaipayan Shikari last updated on 28/Oct/20

$$\underset{x\to \frac{\pi }{3}}{\lim }\frac{sin(x-\frac{\pi }{3})}{1-2cosx}=\frac{1}{2}\frac{sin(x-\frac{\pi }{3})}{cos\frac{\pi }{3}-cosx}$$$$=\frac{1}{2}.\frac{x-\frac{\pi }{3}}{2sin(\frac{x}{2}+\frac{\pi }{6})sin(\frac{x}{2}-\frac{\pi }{6})}=\frac{2}{4}.\frac{x-\frac{\pi }{3}}{sin\frac{\pi }{3}.(x-\frac{\pi }{3})}=\frac{1}{\sqrt{3}}$$

Answered by bobhans last updated on 28/Oct/20

$$\underset{x\to \pi /3}{\lim }\frac{\sin (x-\pi /3)}{(x-\pi /3)}.\frac{(x-\pi /3)}{1-2\cos x}=$$$$[note\underset{x\to \pi /3}{\lim }\frac{\sin (x-\pi /3)}{x-\pi /3}=1]$$$$\underset{x\to \pi /3}{\lim }\frac{x-\pi /3}{1-2\cos x}=\underset{X\to 0}{\lim }\frac{X}{1-2\cos (X+\frac{\pi }{3})}$$$$\underset{X\to 0}{\lim }\frac{X}{1-2(\frac{1}{2}\cos X-\frac{\sqrt{3}}{2}\sin X)}=$$$$\underset{X\to 0}{\lim }\frac{X}{1-\cos X+\sqrt{3}\sin X}=\underset{X\to 0}{\lim }\frac{1}{\frac{1-\cos X}{X}+\frac{\sqrt{3}\sin X}{X}}$$$$=\underset{X\to 0}{\lim }\frac{1}{\frac{2\sin {}^2\left(\frac{X}{2}\right)}{X}+\frac{\sqrt{3}\sin X}{X}}=\frac{1}{0+\sqrt{3}}=\frac{1}{\sqrt{3}}.$$$$$$

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