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Question Number 90210 by manuel__ last updated on 22/Apr/20

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

Commented by mathmax by abdo last updated on 22/Apr/20

$$f\left(x\right)=\frac{2cosx-1}{3x-\pi }=\frac{2cosx-1}{3(x-\frac{\pi }{3})}wedothechangementx-\frac{\pi }{3}=t\Rightarrow$$$$(x\to \frac{\pi }{3}\iff t\to 0)andf\left(x\right)=\frac{2cos(t+\frac{\pi }{3})-1}{3t}=g\left(t\right)$$$$=\frac{2(costcos\left(\frac{\pi }{3}\right)-sintsin\left(\frac{\pi }{3}\right))-1}{3t}$$$$=\frac{cost-\sqrt{3}sint-1}{3t}=-\frac{1-cost}{3t}-\frac{\sqrt{3}}{3}\frac{sint}{t}$$$$1-cost\sim \frac{t^2}{2}and\frac{sint}{t}\sim 1\Rightarrow g\left(t\right)\sim -\frac{1}{3t}\times \frac{t^2}{2}=-\frac{t}{6}\to 0$$$$\Rightarrow {lim}_{t\to 0}g\left(t\right)=-\frac{\sqrt{3}}{3}={lim}_{x\to \frac{\pi }{3}}f\left(x\right)$$

Answered by john santu last updated on 22/Apr/20

$$L^{\prime }hopital$$$$\underset{x\to \frac{\pi }{3}}{\lim }\frac{2\sin x}{-3}=-\frac{2}{3}\times \sin \left(\frac{\pi }{3}\right)$$$$=-\frac{2}{3}\times \frac{\sqrt{3}}{2}=-\frac{\sqrt{3}}{3}$$

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