Word Count : 199

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Let's evaluate the given limit manually: \[ \lim\limits_{x\to\frac{\pi}{6}}\frac{\sqrt{3}\sin x - \cos x}{x - \frac{\pi}{6}} \] ### Step 1: Check for Indeterminate Form Substituting \( x = \frac{\pi}{6} \): \[ \sqrt{3} \sin \frac{\pi}{6} - \cos \frac{\pi}{6} \] Using known values: \[ \sin \frac{\pi}{6} = \frac{1}{2}, \quad \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} __________ ____ _____ ______ ___ ______ _______.
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