e^tanx-e^x是X的几阶无穷小(X->0)?

高阶无穷小

y =

exp(tan(x))-exp(x)


>> taylor(y,0,5)

ans =

1/3*x^3+1/3*x^4


>> taylor(y,0,10)

ans =

1/3*x^3+1/3*x^4+3/10*x^5+11/45*x^6+479/2520*x^7+127/840*x^8+347/3024*x^9

e^tanx－e^x＝e^x×[e^(tanx-x)－1]x→0时,e^(tanx-x)－1等价于tanx－x,设tanx－x是x的k阶无穷小,则lim(x→0) (tanx-x)/x^k存在且非零,由洛必达法则lim(x→0) (tanx-x)/x^k＝lim(x→0) (sinx-xcosx)/(cosx×x^k)＝li...

e^x=1+x+x^2/2+x^3/6+o(x^4)
tanx=x+1/3*x^3+2/15*x^5+o(x^7)
e^tanx=[1+x+x^2+x^3/6+o(x^4)]*[1+1/3*x^3+o(x^6)]=1+x+x^2+x^3/6+1/3*x^3+o(x^4)
所以e^tanx-e^x=1/3*x^3+o(x^4)
所以e^tanx-e^x是x的三阶无穷小