Question Number 105106 by bemath last updated on 26/Jul/20

lim_(x→0) ((sin (πcos ^2 x))/(3x^2 )) ?

Answered by bramlex last updated on 26/Jul/20

lim_(x→0) ((sin (π cos ^2 x))/(3x^2 )) =   lim_(x→0) ((−2πcos xsin x. cos (πcos ^2 x))/(6x))  lim_(x→0) ((−πsin (2x).cos (πcos ^2 x))/(6x))  = lim_(x→0) {−πcos (πcos ^2 x)}.lim_(x→0) ((sin (2x))/(6x))  = (π/3) ▲

Answered by OlafThorendsen last updated on 26/Jul/20

lim_(x→0) ((sin(π(1−(x^2 /2))^2 ))/(3x^2 ))  lim_(x→0) ((sin(π(1−x^2 )))/(3x^2 ))  lim_(x→0) ((sin(πx^2 ))/(3x^2 ))  lim_(x→0) (π/3).((sin(πx^2 ))/(πx^2 ))  lim_(X→0) (π/3).((sinX)/X) = (π/3)

Answered by mathmax by abdo last updated on 26/Jul/20

let f(x) =((sin(πcos^2 x))/(3x^2 ))  we have sin(πcos^2 x) =sin(π×((1+cos(2x))/2))  =sin((π/2) +(π/2)cos(2x)) =cos((π/2)cos(2x)) ∼cos((π/2)(1−2x^2 )) =sin(πx^2 ) ⇒  f(x) ∼ ((sin(πx^2 ))/(3x^2 )) ∼((πx^2 )/(3x^2 )) =(π/3) ⇒lim_(x→0)   f(x) =(π/3)