lim-x-pi-3-1-2cosx-sin-x-pi-3-

Question Number 169147 by mathlove last updated on 25/Apr/22

\lim_{x \to \frac{π}{3}} \frac{1 - 2 c o s x}{s i n (x - \frac{π}{3})} = ?

Commented by cortano1 last updated on 25/Apr/22

l e t x - \frac{π}{3} = h

\lim_{h \to 0} \frac{1 - 2 \cos (\frac{π}{3} + h)}{\sin h}

= \lim_{h \to 0} \frac{2 \sin (\frac{π}{3} + h)}{\cos h} = 2 \times \frac{1}{2} \sqrt{3} = \sqrt{3}

Commented by infinityaction last updated on 25/Apr/22

p = \lim_{x \to \frac{π}{3}} 2 (\frac{\cos \frac{π}{3} - \cos x}{\sin (x - \frac{π}{3})})

p = \underset{x \to \frac{π}{3}}{\lim 2} (\frac{2 \sin (\frac{x + π / 3}{2}) \sin (\frac{x - π / 3}{2})}{2 \sin (\frac{x - π / 3}{2}) \cos (\frac{x - π / 3}{2})})

p = 2 \sin (\frac{2 π}{3})

p = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}