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Question Number 105157 by bramlex last updated on 26/Jul/20

$$\underset{x\to \frac{\pi }{6}}{\lim }\frac{\cos {}^2\left(\frac{3x}{2}\right)-\sin x}{\sin x+\sqrt{3}\cos x-2}?$$

Answered by bramlex last updated on 26/Jul/20

$$\cos {}^2\left(\frac{3x}{2}\right)=\frac{1}{2}(\cos 3x+1)$$$$\underset{x\to \pi /6}{\lim }\frac{\frac{1}{2}(\cos 3x+1)-\sin x}{\sin x+\sqrt{3}\cos x-2}$$$$\underset{x\to \pi /6}{\lim }\frac{\frac{1}{2}(-3\sin 3x)-\cos x}{\cos x-\sqrt{3}\sin x}$$$$=\frac{\frac{3}{2}+\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}}=\mathrm{\infty }\left(DNE\right)$$

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