求极限x趋于0时(1-cos(x/2))x / (tan x-sin x)关键是答案中有一步化简看不懂cos x*x*(1-cos(x/2)) / sinx(1-cos x)=1-cos(x/2) / 1-cos x
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求极限x趋于0时(1-cos(x/2))x / (tan x-sin x)关键是答案中有一步化简看不懂cos x*x*(1-cos(x/2)) / sinx(1-cos x)=1-cos(x/2) / 1-cos x
求极限x趋于0时(1-cos(x/2))x / (tan x-sin x)
关键是答案中有一步化简看不懂
cos x*x*(1-cos(x/2)) / sinx(1-cos x)
=1-cos(x/2) / 1-cos x
求极限x趋于0时(1-cos(x/2))x / (tan x-sin x)关键是答案中有一步化简看不懂cos x*x*(1-cos(x/2)) / sinx(1-cos x)=1-cos(x/2) / 1-cos x
答案见图片
lim(x->0)[(1-cos(x/2))x] '/ [(tanx-sinx)]'
=lim(x->0) [(1/2)sin(x/2)x+(1-cos(x/2)]/(1/cosx^2-cosx)
=lim(x->0)[ [(1/2)sin(x/2)x+(1-cos(x/2)]cosx^2 ]' / [ (1-cosx^3) ]'
=lim(x->0)[(1/4)cos...
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lim(x->0)[(1-cos(x/2))x] '/ [(tanx-sinx)]'
=lim(x->0) [(1/2)sin(x/2)x+(1-cos(x/2)]/(1/cosx^2-cosx)
=lim(x->0)[ [(1/2)sin(x/2)x+(1-cos(x/2)]cosx^2 ]' / [ (1-cosx^3) ]'
=lim(x->0)[(1/4)cos(x/2)xcosx^2 +[(1/2)sin(x/2)x+(1-cos(x/2)](-2sinxcosx) ] /[-3cosx^2]
=0
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对于cos x*x*(1-cos(x/2)) / sinx(1-cos x)
x趋于0时，cosx趋于1，x/sinx趋于1
所以=1-cos(x/2) / 1-cos x
=2sin²(x/4)/2sin²(x/2)
={sin²(x/4)/(x/4)...
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对于cos x*x*(1-cos(x/2)) / sinx(1-cos x)
x趋于0时，cosx趋于1，x/sinx趋于1
所以=1-cos(x/2) / 1-cos x
=2sin²(x/4)/2sin²(x/2)
={sin²(x/4)/(x/4)² }*{(x/2)²/sin²(x/2)} *1/4
—›1/4 (x—›0)
关键是lim(x->0) sinx/x=1的运用
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1/4