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Question Number 70596 by mathmax by abdo last updated on 06/Oct/19

caoculate lim_(x→0) ((cos(sin(x^2 ))−1)/x^2 )

caoculatelimx→0cos(sin(x2))−1x2

Commented by mathmax by abdo last updated on 06/Oct/19

we have sin(x^2 )∼ x^2 ( x∈v(0)) ⇒cos(sinx^2 )∼cos(x^2 )∼1−(x^4 /2) ⇒cos(sin(x^2 ))−1 ∼−(x^4 /2) ⇒((cos(sin(x^2 ))−1)/x^2 ) ∼−(x^2 /2) ⇒ lim_(x→0) ((cos(sin(x^2 ))−1)/x^2 )=0

wehavesin(x2)∼x2(x∈v(0))⇒cos(sinx2)∼cos(x2)∼1−x42⇒cos(sin(x2))−1∼−x42⇒cos(sin(x2))−1x2∼−x22⇒limx→0cos(sin(x2))−1x2=0

Answered by Henri Boucatchou last updated on 06/Oct/19

lim_(x→0) ((cos(sinx^2 )−1)/x^2 ) = lim_(X→0) ((cos(sinX)−cos(sin0))/(X−0)) = cos(sinX)]′_((X=0)) = −cosXsin(sinX)]_((X=0)) = 0

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