Question

求解一道极限

Answer 1

x->0

ln(1+x) = x -(1/2)x^2+(1/3)x^3 +o(x^3)

ln(1+x)/x = 1 -(1/2)x+(1/3)x^2 +o(x^2) 

ln[ln(1+x)/x] 

=ln[ 1 -(1/2)x+(1/3)x^2 +o(x^2) ]

=[-(1/2)x+(1/3)x^2] -(1/2)[-(1/2)x+(1/3)x^2]^2 +o(x^2)

=[-(1/2)x+(1/3)x^2] -(1/2)[(1/4)x^2+o(x^2)] +o(x^2)

=-(1/2)x  +(5/24)x^2 +o(x^2)

[ ln(1+x)/x]^(1/x)

=e^{ ln[ ln(1+x)/x] /x } 

=e^{ [-(1/2)x  +(5/24)x^2 +o(x^2) ] /x } 

=e^ [-1/2 +(5/24)x +o(x) ] 

=(1/√e) e^{(5/24)x +o(x) }

=(1/√e) .[ 1+ (5/24)x +o(x)  ] 

[ ln(1+x)/x]^(1/x) - 1/√e = [5/(24√e)] x +o(x)

lim(x->0) { [ ln(1+x)/x]^(1/x) - 1/√e }/x

=lim(x->0) [5/(24√e)] x/x

=5/(24√e)

Answer 2

追问

用重要极限那边有问题吧？不是指数要和括号里面的互为倒数吗

追答

不需要