Important Equivalent Infinitesimals

Important Equivalent Infinitesimals

As $$x\to0$$:

$$\sin x\sim x$$

$$\tan x\sim x$$

$$\ln(1+x)\sim x$$

$$e^x-1\sim x$$

$$1-\cos x\sim\frac{x^2}{2}$$

Terminology

- $$f(x)\sim g(x)$$ → $$\lim_{x\to0}\frac{f(x)}{g(x)}=1$$.

Usage

- Used to replace complicated expressions by equivalent simpler expressions while evaluating limits.