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Question Number 141885 by mnjuly1970 last updated on 24/May/21
           easy  question:      if   lim_(x→0) ((1−cos(1−cos(1−cos(x))))/x^8 ) =2^( a)           then   a=??
easyquestion:iflimx01cos(1cos(1cos(x)))x8=2athena=??
Answered by Dwaipayan Shikari last updated on 24/May/21
lim_(x→0) ((1−cos(1−cos(1−cos(x))))/x^8 )=((1−cos(1−cos(x^2 /2)))/x^8 )=((1−cos(2sin^2 (x^2 /4)))/x^8 )  =((1−cos((x^4 /8)))/x^8 )=((2sin^2 ((x^4 /(16))))/x^8 )=(1/(128))=2^(−7)
limx01cos(1cos(1cos(x)))x8=1cos(1cosx22)x8=1cos(2sin2x24)x8=1cos(x48)x8=2sin2(x416)x8=1128=27
Commented by mnjuly1970 last updated on 24/May/21
thanks alot..
thanksalot..
Answered by iloveisrael last updated on 24/May/21
 lim_(x→0)  ((2sin^2 (((1−cos (1−cos x))/2)))/x^8 )  = lim_(x→0)  ((2sin^2 (((2sin^2 (((1−cos x)/2)))/2)))/x^8 )  = lim_(x→0)  ((2sin^2 (((2sin^2 (((2sin^2 ((x/2)))/2)))/2)))/x^8 )  = lim_(x→0)  ((2.sin^2 ((x^4 /2^4 )))/x^8 )   = lim_(x→0)  ((2.((x^8 /2^8 )))/x^8 ) = (1/2^7 )
limx02sin2(1cos(1cosx)2)x8=limx02sin2(2sin2(1cosx2)2)x8=limx02sin2(2sin2(2sin2(x2)2)2)x8=limx02.sin2(x424)x8=limx02.(x828)x8=127
Answered by mathmax by abdo last updated on 24/May/21
1−cosx∼(x^2 /2) ⇒cos(1−cosx)∼cos((x^2 /2))∼1−(x^4 /8) ⇒  1−cos(1−cosx)∼(x^4 /8) ⇒cos(1−cos(1−cosx))∼cos((x^4 /8))  ∼1−(x^8 /(2.64)) ⇒1−cos(1−cos(1−cosx))∼(x^8 /2^7 ) ⇒  lim_(x→0)    ((1−cos(1−cos(1−cosx)))/x^8 )=(1/(128))=2^(−7)  ⇒a=−7
1cosxx22cos(1cosx)cos(x22)1x481cos(1cosx)x48cos(1cos(1cosx))cos(x48)1x82.641cos(1cos(1cosx))x827limx01cos(1cos(1cosx))x8=1128=27a=7

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