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Question Number 216032 by efronzo1 last updated on 26/Jan/25

  lim_(x→0)  ((1−cos x (√(cos 2x)))/x^2 ) =?

limx01cosxcos2xx2=?

Answered by mr W last updated on 26/Jan/25

cos x∼1−(x^2 /2)  (√(cos 2x))∼1−x^2   ((1−cos x(√(cos 2x)))/x^2 )  ∼((1−(1−(x^2 /2))(1−x^2 ))/x^2 )  ∼((((1/2)+1)x^2 )/x^2 )  →(3/2) ✓

cosx1x22cos2x1x21cosxcos2xx21(1x22)(1x2)x2(12+1)x2x232

Answered by golsendro last updated on 26/Jan/25

= lim_(x→0)  ((1−cos x)/x^2 ) −lim_(x→0) ((((√(cos x))−1)/x^2 ))cos x   = (1/2) −lim_(x→0)  ((cos x−1)/(x^2 ((√(cos x))+1 )))   = (1/2) + 1 = (3/2)

=limx01cosxx2limx0(cosx1x2)cosx=12limx0cosx1x2(cosx+1)=12+1=32

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