Key to "Limit limx→π3sin⁡(x−π3)1−2cos⁡{x}"

Dominique Pace

Answered question

2022-03-22

Limit

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

Payten Reese

Beginner2022-03-23Added 10 answers

Set

x - \frac{π}{3} = t

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

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

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

Now it should be clear. The limit according to me is

\frac{1}{\sqrt{3}}

Kingston Lowery

Beginner2022-03-24Added 10 answers

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

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