lim(x->0) { [ln(x+√(1+x^2)) ]^2 +e^(-x^2) -1 }/x^4
用泰勒公式来解决问题
分母是4阶
分子要展开到4阶
x->0
ln[x+√(1+x^2)]= x -(1/6)x^3 +o(x^3)
{ln[x+√(1+x^2)]}^2
=[ x -(1/6)x^3 +o(x^3)]^2
=x^2 - (1/3)x^4 +o(x^4)
e^(-x^2) -1 =-x^2 + (1/2)x^4 +o(x^4)
{ln[x+√(1+x^2)]}^2 +e^(-x^2) -1 = (1/6)x^4 +o(x^4)
//
lim(x->0) { [ln(x+√(1+x^2)) ]^2 +e^(-x^2) -1 }/x^4
= lim(x->0) (1/6)x^4/x^4
=1/6