Without-L-Hopital-rule-lim-x-pi-3-sin-x-pi-3-1-2cos-x-

Question Number 119961 by bobhans last updated on 28/Oct/20

W i t h o u t L^{'} H o p i t a l r u l e

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

Answered by bramlexs22 last updated on 28/Oct/20

Answered by Dwaipayan Shikari last updated on 28/Oct/20

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

= \frac{1}{2} . \frac{x - \frac{π}{3}}{2 s i n (\frac{x}{2} + \frac{π}{6}) s i n (\frac{x}{2} - \frac{π}{6})} = \frac{2}{4} . \frac{x - \frac{π}{3}}{s i n \frac{π}{3} . (x - \frac{π}{3})} = \frac{1}{\sqrt{3}}

Answered by bobhans last updated on 28/Oct/20

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

[n o t e \lim_{x \to π / 3} \frac{\sin (x - π / 3)}{x - π / 3} = 1]

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

\lim_{X \to 0} \frac{X}{1 - 2 (\frac{1}{2} \cos X - \frac{\sqrt{3}}{2} \sin X)} =

\lim_{X \to 0} \frac{X}{1 - \cos X + \sqrt{3} \sin X} = \lim_{X \to 0} \frac{1}{\frac{1 - \cos X}{X} + \frac{\sqrt{3} \sin X}{X}}

= \lim_{X \to 0} \frac{1}{\frac{2 \sin^{2} (\frac{X}{2})}{X} + \frac{\sqrt{3} \sin X}{X}} = \frac{1}{0 + \sqrt{3}} = \frac{1}{\sqrt{3}} .

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