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

    ((♥JS♥)/(≤°≡°≤))   lim_(x→π/6) ((2−(√3) cos x−sin x)/((6x−π)^2 )) ?

JS°°limxπ/623cosxsinx(6xπ)2?

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

good all sir...

goodallsir...

Answered by bemath last updated on 15/Aug/20

     ((BeMath)/(★°•°★))   lim_(x→π/6) ((2−(√3) cos x−sin x)/((6x−π)^2 )) ?  set x = (π/6)+w ⇒6x=π+6w  lim_(w→0)  ((2−{(√3) cos (w+(π/6))+sin (w+(π/6))})/((6w)^2 ))=  lim_(w→0) ((2−2cos (w+(π/6)−(π/6)))/(36w^2 ))=  lim_(w→0) ((2(1−cos w))/(36w^2 )) = lim_(w→0) ((2(2sin^2 ((w/2))))/(36w^2 ))  = 4×(1/4)×(1/(36)) = (1/(36))

BeMath°°limxπ/623cosxsinx(6xπ)2?setx=π6+w6x=π+6wlimw02{3cos(w+π6)+sin(w+π6)}(6w)2=limw022cos(w+π6π6)36w2=limw02(1cosw)36w2=limw02(2sin2(w2))36w2=4×14×136=136

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

Put x−(π/6)=t we have 6x=6t+π   lim_(x→(π/6)) ((2−(√3) cos x−sin x)/((6x−π)^2 )) =lim_(t→0) ((2−(√3)cos(t+(π/6))−sin(t+(π/6)))/(36t^2 ))  =lim_(t→0) ((2−(√3) [((√3)/( 2)) cost−(1/2)sint)−((√3)/2)sint−(1/2)cost)/(36t^2 ))  lim(_(t→0) ((2−2cost)/(36t^2 )))=lim(_(t→0) ((4sin^2 (t/2))/(36t^2 )))=lim(_(t→0) ((sin(t/2))/(t/2)))^2 ×(1/(36))  =(1/(36))

Putxπ6=twehave6x=6t+πlimxπ623cosxsinx(6xπ)2=limt023cos(t+π6)sin(t+π6)36t2=limt023[32cost12sint)32sint12cost36t2Extra \left or missing \right=136

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

lim_(x→(π/6)) (((√3)sinx−cosx)/(2.6.(6x−π)))=(((√3)cosx+sinx)/(2.6.6))=(((3/2)+(1/2))/(2.6.6))=(1/(36))

limxπ63sinxcosx2.6.(6xπ)=3cosx+sinx2.6.6=32+122.6.6=136

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

let f(x) =((2−(√3)cosx−sinx)/(36(x−(π/6))^2 ))  changement x−(π/6)=t give  f(x)=((2−(√3)cos(t+(π/6))−sin(t+(π/6)))/(36t^2 ))  (x→(π/6)⇒t→0)  =((2−(√3){((√3)/2)cost−(1/2)sint}−(((√3)/2)sint +(1/2)cost))/(36t^2 ))  =((2−2cost )/(36t^2 ))=((1−cost)/(18t^2 )) ∼((t^2 /2)/(18t^2 )) =(1/(36)) (t→0) ⇒lim_(x→(π/6))   f(x)=(1/(36))

letf(x)=23cosxsinx36(xπ6)2changementxπ6=tgivef(x)=23cos(t+π6)sin(t+π6)36t2(xπ6t0)=23{32cost12sint}(32sint+12cost)36t2=22cost36t2=1cost18t2t2218t2=136(t0)limxπ6f(x)=136

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