Question

数学同阶无穷小X趋向于0时~`e^tanx-e^x 与x^n 是同阶无穷小,问n=?

Answer 1

需要利用泰勒级数展开

当x=0时,

tanx'=1/(cosx^2)=1

tanx''=2sinx/(cosx^2)=0

tanx'''=2

忽略高阶,所以tanx=x+2/(3!)x^3=x+(x^3)/3

同时忽略高阶,

e^x=1+x+x^2/(2!)+x^3/(3!)

而e^tanx=1+tanx+tanx^2/(2!)+x^3/(3!)

=1+(x+x^3/3)+1/2(x+x^3/3)^2+1/6(x+x^3)^3

=1+x+(1/3)x^3+(1/2)x^2+1/6x^3此处忽略了4阶及其更高的阶次

所以：e^tanx-e^x=(1/3)x^3

故与x^3同阶

n=3