做题时常用的等价无穷小有哪些?
当 x→0x→0 时
(01) sinx∽xsinx∽x
(02) tanx∽xtanx∽x
(03) arcsinx∽xarcsinx∽x
(04) arctanx∽xarctanx∽x
(05) ln(1+x)∽xln(1+x)∽x
(06) ex−1∽xex−1∽x
(07) 1−cosx∽12x21−cosx∽12x2
(08) x−ln(1+x)∽12x2x−ln(1+x)∽12x2
(09) tanx−sinx∽12x3tanx−sinx∽12x3
(10) arcsinx−arctanx∽12x3arcsinx−arctanx∽12x3
(11) tanx−x∽13x3tanx−x∽13x3
(12) x−arctanx∽13x3x−arctanx∽13x3
(13) x−sinx∽16x3x−sinx∽16x3
(14) (1+a)x−1∽ax(1+a)x−1∽ax
(15) ax−1∽lna×x
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);
还有吗?记得老师写了不少,我都没记住