Evaluate-lim-x-pi-6-3-sin-x-cos-x-x-pi-6-

Question Number 184796 by Spillover last updated on 11/Jan/23

E v a l u a t e

\lim_{x \to \frac{π}{6}} \frac{\sqrt{3} \sin x - \cos x}{x - \frac{π}{6}}

Commented by MJS_new last updated on 11/Jan/23

2

Commented by MJS_new last updated on 11/Jan/23

these are all easy with l^{'} H \hat{o p i t a l}

Answered by aba last updated on 12/Jan/23

let t = x - \frac{π}{6} \Rightarrow x = t + \frac{π}{6}

\lim_{x \to \frac{π}{6}} \frac{\sqrt{3} \sin (x) - \cos (x)}{x - \frac{π}{6}} = \lim_{t \to 0} \frac{\sqrt{3} \sin (t + \frac{π}{6}) - \cos (t + \frac{π}{6})}{t}

= \lim_{t \to 0} \frac{\sqrt{3} (\sin (t) \cos (\frac{π}{6}) + \cos (t) \sin (\frac{π}{6})) - (\cos (t) \cos (\frac{π}{6}) - \sin (t) \sin (\frac{π}{6}))}{t}

= \lim_{t \to 0} \frac{\sqrt{3} (\sqrt{3} \sin (t) + \cos (t)) - (\sqrt{3} \cos (t) - \sin (t))}{2 t}

= \lim_{t \to 0} \frac{3 \sin (t) + \sin (t) + \sqrt{3} \sin (t) - \sqrt{3} \cos (t)}{2 t}

= \lim_{t \to 0} \frac{4 \sin (t)}{2 t}

= 2

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