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Question Number 171235 by thean last updated on 10/Jun/22

Commented by eman_64 last updated on 10/Jun/22

Commented by cortano1 last updated on 11/Jun/22

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

Answered by Mathspace last updated on 10/Jun/22

$$f\left(x\right)=\frac{\sqrt{3}cosx-sinx}{3(x-\frac{\pi }{3})}$$$$wedothechangementx-\frac{\pi }{3}=t$$$$(x\to \frac{\pi }{3}\iff t\to 0)and$$$$f\left(x\right)=f(t+\frac{\pi }{3})=\frac{\sqrt{3}cos(t+\frac{\pi }{3})-sin(t+\frac{\pi }{3})}{3t}$$$$=\frac{\sqrt{3}\{\frac{1}{2}cost-\frac{\sqrt{3}}{2}sint\}-\{\frac{1}{2}sint+\frac{\sqrt{3}}{2}cost\}}{3t}$$$$=\frac{\frac{\sqrt{3}}{2}cost-\frac{3}{2}sint-\frac{1}{2}sint-\frac{\sqrt{3}}{2}cost}{3t}$$$$=\frac{-2sint}{3t}\Rightarrow$$$${lim}_{t\to 0}\frac{-2sint}{3t}=-\frac{2}{3}{lim}_{t\to 0}\frac{sint}{t}$$$$=-\frac{2}{3}\times 1=-\frac{2}{3}\Rightarrow$$$${lim}_{x\to \frac{\pi }{3}}f\left(x\right)=-\frac{2}{3}$$$$hospitalmethod$$$${lim}_{x\to \frac{\pi }{3}}f\left(x\right)={lim}_{x\to \frac{\pi }{3}}\frac{-\sqrt{3}sinx-cosx}{3}$$$$=\frac{-\sqrt{3}.\frac{\sqrt{3}}{2}-\frac{1}{2}}{3}=\frac{-\frac{3}{2}-\frac{1}{2}}{3}=-\frac{2}{3}$$

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