常用的等价无穷小一般有:
1)x趋向于0时:sinx~x; tanx~x; 1-cosx~(1/2)x^2; arcsinx~x; arctanx~x; (e^x)-1~x;(a^x)-1~xIna(0<a<1或a>1); In(1+x)~x; (1+x)^a~ax+1;(x^m)+(x^n)~x^m(n>m>0); lim(1+x)^(1/x)=e;
2)n趋向于无穷大时:lim[n^(1/n)]=1;lim[a^(1/n)]=1(a>0);lim[1+1/n]^n=e;
3)在必要情况下,采用泰勒展开的高阶等价无穷小:sinx=x-(1/6)x^3+o(x^3);cosx=1-(x^2)/2!+(x^4)/4!+o(x^4);tanx=x+(1/3)x^3+o(x^3);arcsinx=x+(1/6)x^3+o(x^3);arctanx=x-(1/3)x^3+o(x^3);In(1+x)=x-(x^2)/2+(x^3)/3+o(x^3);e^x=1+x+(1/2)x^2+(1/6)x^3+o(x^3);(1+x)^a=1+ax+a(a-1)(x^2)/2+o(x^2);
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