JS-lim-x-pi-6-2-3-cos-x-sin-x-6x-pi-2-

Question Number 108217 by john santu last updated on 15/Aug/20

\frac{♡ J S ♡}{⩽ ° \equiv ° ⩽}

\lim_{x \to π / 6} \frac{2 - \sqrt{3} \cos x - \sin x}{{(6 x - π)}^{2}} ?

Commented by john santu last updated on 15/Aug/20

g o o d a l l s i r \dots

Answered by bemath last updated on 15/Aug/20

\frac{B e M a t h}{★ ° ∙ ° ★}

\lim_{x \to π / 6} \frac{2 - \sqrt{3} \cos x - \sin x}{{(6 x - π)}^{2}} ?

s e t x = \frac{π}{6} + w \Rightarrow 6 x = π + 6 w

\lim_{w \to 0} \frac{2 - {\sqrt{3} \cos (w + \frac{π}{6}) + \sin (w + \frac{π}{6})}}{{(6 w)}^{2}} =

\lim_{w \to 0} \frac{2 - 2 \cos (w + \frac{π}{6} - \frac{π}{6})}{36 w^{2}} =

\lim_{w \to 0} \frac{2 (1 - \cos w)}{36 w^{2}} = \lim_{w \to 0} \frac{2 (2 \sin^{2} (\frac{w}{2}))}{36 w^{2}}

= 4 \times \frac{1}{4} \times \frac{1}{36} = \frac{1}{36}

Answered by 1549442205PVT last updated on 15/Aug/20

Put x - \frac{π}{6} = t we have 6 x = 6 t + π

\lim_{x \to \frac{π}{6}} \frac{2 - \sqrt{3} \cos x - \sin x}{{(6 x - π)}^{2}} = \lim_{t \to 0} \frac{2 - \sqrt{3} \cos (t + \frac{π}{6}) - \sin (t + \frac{π}{6})}{{36 t}^{2}}

= \lim_{t \to 0} \frac{2 - \sqrt{3} [\frac{\sqrt{3}}{2} cost - \frac{1}{2} sint) - \frac{\sqrt{3}}{2} sint - \frac{1}{2} cost}{{36 t}^{2}}

Extra \left or missing \right

= \frac{1}{36}

Answered by Dwaipayan Shikari last updated on 15/Aug/20

\lim_{x \to \frac{π}{6}} \frac{\sqrt{3} s i n x - c o s x}{2.6 . (6 x - π)} = \frac{\sqrt{3} c o s x + s i n x}{2.6.6} = \frac{\frac{3}{2} + \frac{1}{2}}{2.6.6} = \frac{1}{36}

Answered by mathmax by abdo last updated on 15/Aug/20

let f (x) = \frac{2 - \sqrt{3} cosx - sinx}{36 {(x - \frac{π}{6})}^{2}} changement x - \frac{π}{6} = t give

f (x) = \frac{2 - \sqrt{3} \cos (t + \frac{π}{6}) - \sin (t + \frac{π}{6})}{{36 t}^{2}} (x \to \frac{π}{6} \Rightarrow t \to 0)

= \frac{2 - \sqrt{3} {\frac{\sqrt{3}}{2} cost - \frac{1}{2} sint} - (\frac{\sqrt{3}}{2} sint + \frac{1}{2} cost)}{{36 t}^{2}}

= \frac{2 - 2 cost}{{36 t}^{2}} = \frac{1 - cost}{{18 t}^{2}} \sim \frac{\frac{t^{2}}{2}}{{18 t}^{2}} = \frac{1}{36} (t \to 0) \Rightarrow \lim_{x \to \frac{π}{6}} f (x) = \frac{1}{36}