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=?

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)=1tanx''=2sinx/(cosx^2)=0tanx'''=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...