Question

请问一下这个式子怎么解？

Answer 1

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