Infinitesimals

Equivalent Infinitesimals

$\alpha (x)$ and $\beta (x)$ are equivalent infinitesimals, written as $\alpha (x)\sim \beta (x)$ saids that

$$\underset{x\to a}{\lim }\frac{\alpha (x)}{\beta (x)}=1$$

oscillating function

$\alpha$ and $\beta$ should not cross zero “near” $a$, otherwise, the quotient is not defined.

Transitivity of equivalent

As for Product Rule, so the equivalent relation has transitivity:

$$\alpha \sim \beta ,\beta \sim \gamma ⟹\alpha \sim \gamma$$

Proof:

$$\underset{x\to a}{\lim }\frac{\alpha (x)}{\gamma (x)}=\underset{x\to a}{\lim }\frac{\alpha (x)}{\beta (x)}\frac{\beta (x)}{\gamma (x)}=\underset{x\to a}{\lim }\frac{\alpha (x)}{\beta (x)}\underset{x\to a}{\lim }\frac{\beta (x)}{\gamma (x)}=1\cdot 1=1$$

Equivalent Infinitesimal Trick

$$\begin{array}{rl}\underset{x\to a}{\lim }\frac{\alpha (x)}{\beta (x)}& =\underset{x\to a}{\lim }\frac{\tilde{\alpha }(x)}{\beta (x)}\\ \underset{x\to a}{\lim }\frac{\alpha (x)}{\beta (x)}& =\underset{x\to a}{\lim }\frac{\alpha (x)}{\tilde{\beta }(x)}\end{array}$$