8个常用泰勒公式:
sin x = x − 1 6 x 3 + O ( x 3 ) arcsin x = x + 1 6 x 3 + O ( x 3 ) sin x=x-frac{1}{6} x^{3}+Oleft(x^{3}
ight) quad arcsin x=x+frac{1}{6} x^{3}+Oleft(x^{3}
ight)sinx=x−
6
1
x
3
+O(x
3
)arcsinx=x+
6
1
x
3
+O(x
3
)
cos x = 1 − 1 2 x 2 + x 4 4 ! + 0 ( x 4 ) ln ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 + O ( x 3 ) cos x=1-frac{1}{2} x^{2}+frac{x^{4}}{4 !}+0left(x^{4}
ight) quad ln (1+x)=x-frac{1}{2} x^{2}+frac{1}{3} x^{3}+O(x^{3})cosx=1−
2
1
x
2
+
4!
x
4
+0(x
4
)ln(1+x)=x−
2
1
x
2
+
3
1
x
3
+O(x
3
)
tan x = x + 1 3 x 3 + O ( x 3 ) arctan x = x − 1 3 x 3 + O ( x 3 )
an x=x+frac{1}{3} x^{3}+O( x^{3}) quad arctan x=x-frac{1}{3} x^{3}+Oleft(x^{3}
ight)tanx=x+
3
1
x
3
+O(x
3
)arctanx=x−
3
1
x
3
+O(x
3
)
e x = 1 + x + 1 2 x 2 + 1 6 x 3 + 0 ( x 3 ) ( 1 + x ) a = 1 + a x + + a ( a − 1 ) 2 ! x 2 + O ( x 2 ) e^{x}=1+x+frac{1}{2} x^{2}+frac{1}{6} x^{3}+0left(x^{3}
ight) quad(1+x)^{a}=1+a x++frac{a(a-1)}{2 !} x^{2}+Oleft(x^{2}
ight)e
x
=1+x+
2
1
x
2
+
6
1
x
3
+0(x
3
)(1+x)
a
=1+ax++
2!
a(a−1)
x
2
+O(x
2
)
泰勒公式是等号而不是等价，这就使所有函数转化为幂函数，在利用高阶无穷小被低阶吸收的原理，可以秒杀大部分极限题。
