x ∼ sin ⁡ x ∼ tan ⁡ x ∼ arcsin ⁡ x ∼ arctan ⁡ x ∼ ln ⁡ ( 1 + x ) ∼ e x − 1 x \thicksim \sin x \thicksim \tan x \thicksim \arcsin x \thicksim \arctan x \thicksim \ln(1+x) \thicksim e^x-1 x∼sinx∼tanx∼arcsinx∼arctanx∼ln(1+x)∼ex−1

a x − 1 ∼ x ln ⁡ a a^x-1 \thicksim x \ln a ax−1∼xlna

( 1 + x ) α − 1 ∼ α x (1+x)^\alpha - 1 \thicksim \alpha x (1+x)α−1∼αx

x − l n ( 1 + x ) ∼ 1 2 x 2 x-ln(1+x) \thicksim \dfrac{1}{2}x^2 x−ln(1+x)∼21​x2

1 − cos ⁡ x ∼ 1 2 x 2 ,   1 − c o s k x ∼ k 2 x 2 1-\cos x \thicksim \dfrac{1}{2}x^2,\ 1-cos^k x \thicksim \dfrac{k}{2}x^2 1−cosx∼21​x2, 1−coskx∼2k​x2

sin ⁡ x − x ∼ − 1 6 x 3 ,   arcsin ⁡ x − x ∼ 1 6 x 3 \sin x - x \thicksim -\dfrac{1}{6}x^3,\ \arcsin x - x \thicksim \dfrac{1}{6}x^3 sinx−x∼−61​x3, arcsinx−x∼61​x3

tan ⁡ x − x ∼ 1 3 x 3 ,   arctan ⁡ x − x ∼ − 1 3 x 3 \tan x - x \thicksim \dfrac{1}{3}x^3,\ \arctan x - x \thicksim -\dfrac{1}{3}x^3 tanx−x∼31​x3, arctanx−x∼−31​x3

注： A-B型的等价无穷小，可通过泰勒公式推导出来。推导原则是，“幂次最低”原则，即展开到系数不相等的 x x x 的最低次幂。