Examples

1) The relation $f \ll 1$ means that $f$ tends to 0 along $\mathfrak{F}$.

2) When $\alpha$ and $\beta$ are two real numbers such that $\alpha < \beta$ one has $x^\alpha \ll x^\beta$ as $x$ tends to $+\infty$. Similarly, when $m$ and $n$ are two rational integers such that $m < n$ one has $z^m \ll z^n$ as the complex number $z$ tends to $\infty$.

3) As $x$ tends to $+\infty$ one has $x^n \ll e^x$ for every integer $n$ (III, p. 105).

4) In $\mathbf{R}^2$ one has, as $(x, y)$ tends to $(0, 0)$,

The following propositions can be deduced immediately from def. 3: