Question

ln【(1+x)/(1-根号x)】与根号x是等价无穷小吗? 求给出证明.

Answer 1

与根号X是等价无穷小。证：
当x-->0时, ln(1+x)～x ln(1-根号x)～ -根号x
lim[x-->0]{ln[(1+x)/(1-根号x)]/根号x}
=lim[x-->0]{ln(1+x)/根号x-ln(1-根号x)/根号x}
=lim[x-->0][x/根号x-(-根号x)/根号x]
=0-(-1)
=1
得证

Answer 2

(1+x)/(1-√x)
=1 +(√x+x)/(1-√x)
~ 1+(√x+x)
~ 1+√x
ln[(1+x)/(1-√x)]
~ln(1+√x)
~ √x
ie
ln[(1+x)/(1-√x)] 等价于√x