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

Question Number 90210 by manuel__ last updated on 22/Apr/20

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

Commented by mathmax by abdo last updated on 22/Apr/20

f (x) = \frac{2 c o s x - 1}{3 x - π} = \frac{2 c o s x - 1}{3 (x - \frac{π}{3})} w e d o t h e c h a n g e m e n t x - \frac{π}{3} = t \Rightarrow

(x \to \frac{π}{3} \Leftrightarrow t \to 0) a n d f (x) = \frac{2 c o s (t + \frac{π}{3}) - 1}{3 t} = g (t)

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

= \frac{c o s t - \sqrt{3} s i n t - 1}{3 t} = - \frac{1 - c o s t}{3 t} - \frac{\sqrt{3}}{3} \frac{s i n t}{t}

1 - c o s t \sim \frac{t^{2}}{2} a n d \frac{s i n t}{t} \sim 1 \Rightarrow g (t) \sim - \frac{1}{3 t} \times \frac{t^{2}}{2} = - \frac{t}{6} \to 0

\Rightarrow {l i m}_{t \to 0} g (t) = - \frac{\sqrt{3}}{3} = {l i m}_{x \to \frac{π}{3}} f (x)

Answered by john santu last updated on 22/Apr/20

L^{'} h o p i t a l

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

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