常用等价无穷小(常用等价无穷小有哪些)

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

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);