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

Question Number 84492 by john santu last updated on 13/Mar/20

\lim_{x \to \frac{π}{3}} \frac{\sin (x - \frac{π}{3})}{1 - 2 \cos (x)} =

Commented by john santu last updated on 13/Mar/20

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

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

Commented by john santu last updated on 13/Mar/20

other way

\sin (x - \frac{π}{3}) \approx (x - \frac{π}{3})

\cos (x) \approx \frac{1}{2} - \frac{\sqrt{3}}{2} (x - \frac{π}{3})

\lim_{x \to \frac{π}{3}} \frac{x - \frac{π}{3}}{1 - 2 [\frac{1}{2} - \frac{\sqrt{3}}{2} (x - \frac{π}{3})]} =

\lim_{x \to \frac{π}{3}} \frac{(x - \frac{π}{3})}{\sqrt{3} (x - \frac{π}{3})} = \frac{1}{\sqrt{3}} .

Commented by mathmax by abdo last updated on 13/Mar/20

l e t l (x) = \frac{s i n (x - \frac{π}{3})}{1 - 2 c o s x} c h a n g e m e n t x - \frac{π}{3} = t g i v e

l (x) = A (t) = \frac{s i n t}{1 - 2 c o s (t + \frac{π}{3})} = \frac{s i n t}{1 - 2 (c o s t c o s \frac{π}{3} - s i n t s i n (\frac{π}{3}))}

= \frac{s i n t}{1 - 2 (\frac{1}{2} c o s t - \frac{\sqrt{3}}{2} s i n t)} = \frac{s i n t}{1 - c o s t + \sqrt{3} s i n t}

x \to \frac{π}{3} \Leftrightarrow t \to 0 a n d A (t) \sim \frac{t}{\frac{t^{2}}{2} + \sqrt{3} t} = \frac{1}{\frac{t}{2} + \sqrt{3}} \Rightarrow {l i m}_{t \to 0} A (t) = \frac{1}{\sqrt{3}}

\Rightarrow {l i m}_{x \to 0} l (x) = \frac{1}{\sqrt{3}}