Question Number 10374 by ridwan balatif last updated on 06/Feb/17

lim_(x→(π/4)) (((x−(π/4))sin(3x−3(π/4)))/(2(1−sin2x)))=...?

Answered by mrW1 last updated on 06/Feb/17

let u=x−(π/4)  with x→(π/4), u→0  sin 2x=sin (2u+(π/2))=cos (2u)=1−2 sin^2  u    L=lim_(x→(π/4)) (((x−(π/4))sin(3x−3(π/4)))/(2(1−sin2x)))  =lim_(u→0) ((u sin (3u))/(4 sin^2  u))=lim_(u→0)  ((u×(((sin 3u)/(3u)))×3u)/(4×(((sin u)/u))^2 ×u^2 ))  =lim_(u→0) ((3×(((sin 3u)/(3u))))/(4×(((sin u)/u))^2 ))=((3×1)/(4×1^2 ))=(3/4)

Commented byridwan balatif last updated on 06/Feb/17

thank you sir

Answered by arge last updated on 08/Feb/17

por l′hopital,    y=(((x−(x/4))sen(3x−((12+π)/4)))/(2(1−sen2x)))    y′=((2(1−sen2x)[(x−(x/4))cos(3x−((12+π)/4))+sen(3x−((12+π)/4))(1−(1/4))]−(x−(x/4))sen(3x−((12+π)/4))×2(−cos2x))/(4(1−2sen2x+4sen^2 xcos^2 x)))    y′=(A/0)    A=_   −134.65, y′=∞∵∵∵∵∵Rta

Answered by bahmanfeshki last updated on 02/Mar/17

1−sin 2x=(sin x+cos x)^2 =2sin^2  (x+(π/4))  x+(π/4)=t  lim_(t→0)  ((tsin 3t)/(4sin ^2 t))=(3/4)lim_(t→0) (((sin 3t)/(3t))/((((sin t)/t))^2 ))=(3/4)