等价无穷小
sinx∼arcsinx∼tanx∼arctanx∼x
ln(1+x)∼ex−1∼1+x−1−x∼x
1−cosx∼secx−1∼x−ln(1+x)∼21x2
1−cosax∼2ax2
(1+x)a−1∼ax
ax−1∼xlna
loga(1+x)∼lnax
tanx−x∼x−arctanx∼3x3
x−sinx∼arcsinx−x∼6x3
泰勒公式
ex=1+x+2!x2+⋅⋅⋅+n!xn+o(xn)
1−x1=1+x+x2+...+xn+o(xn)
1+x1=1−x+x2+...+(−1)nxn+o(xn)
(1+x)α=1+αx+2!α(α−1)x2+⋅⋅⋅+n!α(α−1)⋅⋅⋅(α−n+1)xn+o(xn)
ln(1+x)=x−21x2+31x3−⋅⋅⋅+(−1)nn+11xn+1+o(xn+1)
cosx=1−2!1x2+4!1x4+o(x4)
sinx=x−3!x3+5!x5+o(x5)
tanx=x+3x3+152x5+o(x5)
arctanx=x−3x3+5x5+o(x5)
arcsinx=x+61x3+403x5+o(x5)
secx=1+21x2+245x4+o(x4)
cscx=x1+61x+3607x3+o(x3)
cotx=x1−31x−451x3+o(x3)
其他
当 x>0 时, ln(1+x)<x;sinx<x
x→0limxsinx=1
x→0lim(1+x)x1=e
x→∞lim(1+x1)x=e
x→0limxax−1=lna
x→∞limnn=1
x→∞limna=1,(a>0)
x→∞limbmxm+bn−1xm−1+⋯+b1x+b0anxn+an−1xn−1+⋯+a1x+a0=⎩⎨⎧bman,0,∞,n=mn<mn>m
x→+∞ 时
lnαx≪xβ≪ax, 其中 α>0,β>0,a>1
x→∞时
lnαn≪nβ≪an≪n!≪nn,其中α>0,β>0,a>1
n→∞lim na1n+a2n+⋯+amn=a