顶🔝

x->0
分子
ln(2-cosx)
~ln(1+(1/2)x^2-(1/24)x^4)
~[(1/2)x^2-(1/24)x^4] -(1/2)[(1/2)x^2-(1/24)x^4]^2
=[(1/2)x^2-(1/24)x^4] -(1/2)[(1/4)x^4+...]
=(1/2)x^2- (1/6)x^4
sinx ~ x -(1/6)x^3
[sinx]^2 ~ [x -(1/6)x^3]^2 ~ x^2 -(1/3)x^4
[1+ (sinx)^2]^(1/3)
~[ 1+x^2 -(1/3)x^4 ]^(1/3)
~ 1 +(1/3)[x^2 -(1/3)x^4] -(1/9)[x^2 -(1/3)x^4]^2
= 1 +(1/3)[x^2 -(1/3)x^4] -(1/9)[x^4+...]
=1 +(1/3)x^2 -(2/9)x^4
2ln(2-cosx) -3[[1+ (sinx)^2]^(1/3) -1]
~2[(1/2)x^2- (1/6)x^4 ] -3[(1/3)x^2 -(2/9)x^4]
=(1/3)x^4
分母
[xln(1+x)]^2~ (x^2)^2 =x^4
//
lim(x->0) { 2ln(2-cosx) -3[[1+ (sinx)^2]^(1/3) -1] }/[xln(1+x)]^2
=lim(x->0) (1/3)x^4/x^4
=1/3

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