lim-x-pi-3-tan-2-x-3-tanx-cos-x-pi-6-

Question Number 142444 by mathdanisur last updated on 31/May/21

\underset{x \to \frac{π}{3}}{l i m} \frac{{t a n}^{2} x - \sqrt{3} t a n x}{c o s (x + \frac{π}{6})} = ?

Answered by bramlexs22 last updated on 31/May/21

\lim_{x \to \frac{π}{3}} \frac{\tan^{2} x - \sqrt{3} \tan x}{\cos (x + \frac{π}{6})} = ?

s e t x = \frac{π}{3} + z a n d z \to 0

\Leftrightarrow \lim_{z \to 0} \frac{\tan (z + \frac{π}{3}) [\tan (z + \frac{π}{3}) - \sqrt{3}]}{\cos (z + \frac{π}{2})}

= - \lim_{z \to 0} \tan (z + \frac{π}{3}) . \lim_{z \to 0} \frac{\tan (z + \frac{π}{3}) - \sqrt{3}}{\sin z}

= - \sqrt{3} . \lim_{z \to 0} \frac{\sec^{2} (z + \frac{π}{3})}{\cos z}

= - \sqrt{3} . \lim_{z \to 0} \frac{1}{\cos^{2} (z + \frac{π}{3}) . \cos z}

= - \sqrt{3} . \frac{1}{(\frac{1}{4}) . 1} = - 4 \sqrt{3}

Commented by mathdanisur last updated on 02/Jun/21

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