lim_(x→π/3) ((cos x−sin (π/6))/((π/6)−(x/2)))=..


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Question Number 55592 by gunawan last updated on 27/Feb/19

$\lim_{x \to π / 3} \frac{\cos x - \sin \frac{π}{6}}{\frac{π}{6} - \frac{x}{2}} = . .$
Commented by maxmathsup by imad last updated on 27/Feb/19
$let A(x)=((cosx−sin((π/6)))/((π/6)−(x/2))) ⇒ A(x)=6((cosx−(1/2))/(π−3x)) =3 ((2cosx−1)/(3((π/3)−x))) =((1−2cosx)/(x−(π/3))) changement x−(π/3) =t give lim_(x→(π/3)) A(x) =lim_(t→0) ((1−2cos(t+(π/3)))/t) =lim_(t→0) ((1−2{(1/2)cost −((√3)/2)sint)})/t) =lim_(t→0) ((1−cost +(√3)sint)/t) =lim_(t→0) ((1−cost)/t) +lim_(t→0) (√3)((sint)/t) but cost ∼1−(t^2 /2) ⇒((1−cost)/t) ∼ (t/2) →0(t→0) also sint ∼t ⇒lim_(t→0) ((sint)/t) =1 ⇒ lim_(x→(π/3)) A(x)=(√3).$
$l e t A (x) = \frac{c o s x - s i n (\frac{π}{6})}{\frac{π}{6} - \frac{x}{2}} \Rightarrow A (x) = 6 \frac{c o s x - \frac{1}{2}}{π - 3 x} = 3 \frac{2 c o s x - 1}{3 (\frac{π}{3} - x)}$ $= \frac{1 - 2 c o s x}{x - \frac{π}{3}} c h a n g e m e n t x - \frac{π}{3} = t g i v e {l i m}_{x \to \frac{π}{3}} A (x)$ $= {l i m}_{t \to 0} \frac{1 - 2 c o s (t + \frac{π}{3})}{t} = {l i m}_{t \to 0} \frac{1 - 2 {\frac{1}{2} c o s t - \frac{\sqrt{3}}{2} s i n t)}}{t}$ $= {l i m}_{t \to 0} \frac{1 - c o s t + \sqrt{3} s i n t}{t} = {l i m}_{t \to 0} \frac{1 - c o s t}{t} + {l i m}_{t \to 0} \sqrt{3} \frac{s i n t}{t}$ $b u t c o s t \sim 1 - \frac{t^{2}}{2} \Rightarrow \frac{1 - c o s t}{t} \sim \frac{t}{2} \to 0 (t \to 0) a l s o s i n t \sim t \Rightarrow {l i m}_{t \to 0} \frac{s i n t}{t} = 1 \Rightarrow$ ${l i m}_{x \to \frac{π}{3}} A (x) = \sqrt{3} .$
	Answered by kaivan.ahmadi last updated on 27/Feb/19

	$h o p i t a l$ $l i \underset{x \to \frac{π}{3}}{m} \frac{- s i n x}{- \frac{1}{2}} = l i \underset{x \to \frac{π}{3}}{m} 2 s i n x = 2 s i n \frac{π}{3} = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}$
	Answered by math1967 last updated on 27/Feb/19

	$\lim_{x \to \frac{π}{3}} \frac{cosx - \cos (\frac{π}{2} - \frac{π}{6})}{\frac{π}{6} - \frac{x}{2}}$ $\lim_{x \to \frac{π}{3}} \frac{2 \sin (\frac{π}{6} - \frac{x}{2}) \sin (\frac{x}{2} + \frac{π}{6})}{\frac{π}{6} - \frac{x}{2}} ★$ $2 \times 1 \times \sin (\frac{π}{6} + \frac{π}{6}) = 2 \times 1 \times \frac{\sqrt{3}}{2} = \sqrt{3}$ $★ \lim_{(\frac{π}{6} - \frac{x}{2}) \to 0} \frac{\sin (\frac{π}{6} - \frac{x}{2})}{(\frac{π}{6} - \frac{x}{2})} = 1$ $∵ x \to \frac{π}{3} ∴ \frac{x}{2} \to \frac{π}{6} ∴ (\frac{π}{6} - \frac{x}{2}) \to 0$
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