搜狗做题网求X趋于0时 lim[1/ln(1+根号下（1+x平方））-1/ln(1+x)]
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求X趋于0时 lim[1/ln(1+根号下（1+x平方））-1/ln(1+x)]
![求X趋于0时 lim[1/ln(1+根号下（1+x平方））-1/ln(1+x)]](/uploads/image/z/216987-51-7.jpg?t=%E6%B1%82X%E8%B6%8B%E4%BA%8E0%E6%97%B6+lim%5B1%2Fln%281%2B%E6%A0%B9%E5%8F%B7%E4%B8%8B%EF%BC%881%2Bx%E5%B9%B3%E6%96%B9%EF%BC%89%EF%BC%89-1%2Fln%281%2Bx%29%5D)
设y=ln(x+√(1+x^2)),y'=1/√(1+x^2)=1+x^2/2+o(x^2)
y=x+o(x^2),ln(1+x)=x-x^2/2+o(x^2)
原极限=lim(ln(1+x)-ln(x+√(1+x^2))/ln(1+x)ln(x+√(1+x^2))
=lim(x+o(x^2))-(x-x^2/2+o(x^2))/(x+o(x^2))(x-x^2/2+o(x^2))
=lim(x^2/2+o(x^2))/(x^2+o(x^2))
=1/2
y=x+o(x^2),ln(1+x)=x-x^2/2+o(x^2)
原极限=lim(ln(1+x)-ln(x+√(1+x^2))/ln(1+x)ln(x+√(1+x^2))
=lim(x+o(x^2))-(x-x^2/2+o(x^2))/(x+o(x^2))(x-x^2/2+o(x^2))
=lim(x^2/2+o(x^2))/(x^2+o(x^2))
=1/2
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