求lim arctanx-x/ ln(1+3x+2x^3),x趋于0?

问题描述:

求lim arctanx-x/ ln(1+3x+2x^3),x趋于0?

0.
证明:
arctanx-x=-x^3/3+x^5/5+o(x^5),ln(1+3x+2x^3)=3x+2x^3-9x^2+o(x^2).
则原式=(-x^3/3+x^5/5+o(x^5))/(3x-9x^2+2x^3)=(-x^2)(-1/3+x^2/5)/(3-9x+2x^2)=0.