Question Number 148328 by liberty last updated on 27/Jul/21

lim_(x→0) ((sin 2x+2sin ^2 x−2sin x)/(cos x−cos ^2 x)) =?

Answered by EDWIN88 last updated on 27/Jul/21

Answered by mathmax by abdo last updated on 27/Jul/21

let f(x)=((sin(2x)+2sin^2 (x)−2sinx)/(cosx−cos^2 x)) we have sin(2x)∼2x ,sin^2 x∼x^2 cosx−cos^2 x=cosx(1−cosx)∼(1−(x^2 /2)).(x^2 /2)=(x^2 /2)−(x^4 /4) ⇒ f(x)∼((2x+2x^2 −2x)/((x^2 /2)−(x^4 /4)))=(2/((1/2)−(x^2 /4))) ⇒lim_(x→0) f(x)=4