Question

X趋向0 求(e^tanx-e^sinx)/((1-cosx)ln(1+x))的极限

Answer 1

由e^x=1+x+o(x)

又sinx=x-x^3/6+o(x^3),tanx=x+x^3/3+o(x^3)

所以e^tanx-e^sinx=(1+tanx+o(tanx))-(1+sinx+o(sinx))

=tanx-sinx+o(x^3)

=x^3/2+o(x^3)

ln(x+1)=x+o(x)

1-cosx=x^2/2+o(x^2)

所以(1-cosx)ln(1+x)=x^3/2+o(x^3)

lim[x→0](e^tanx-e^sinx)/((1-cosx)ln(1+x))

=lim[x→0](x^3/2)/(x^3/2)

=1