无穷小替换

x → 0 x\rightarrow 0 x0

  • x ∼ sin ⁡ x ∼ tan ⁡ x ∼ arcsin ⁡ x ∼ arctan ⁡ x ∼ l n ( 1 + x ) ∼ e x − 1 x\sim \sin x \sim \tan x \sim \arcsin x \sim \arctan x \sim ln(1+x) \sim e^x-1 xsinxtanxarcsinxarctanxln(1+x)ex1
  • 1 − cos ⁡ x ∼ 1 2 x 2 1 − c o s a x ∼ a 2 x 2 1-\cos x \sim \frac12x^2 \quad 1-cos^a x \sim \frac a2 x^2 1cosx21x21cosax2ax2
  • 1 + x − 1 ∼ 1 2 x ( 1 + x ) a − 1 ∼ a x \sqrt{1+x}-1\sim \frac12x \quad (1+x)^a-1 \sim ax 1+x 121x(1+x)a1ax

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