Infinitesimals, Big O, Small o

Infinitesimal

A function $f(x)$ is said to be an infinitesimal as $x \to c$, if

$$\begin{equation*} \lim_{x \to c} f(x) = 0. \end{equation*}$$

A function $f(x)$ is said to be a bounded quantity as $x \to c$ if there exists $M > 0$ and a deleted neighborhood $W$ of $c$ such that for all $x \in W \cap I$,

$$\begin{equation*} |f(x)| \le M. \end{equation*}$$

Big O

When $x \to c$, if $f(x) = O(g(x))$, it means:

There exists $M > 0$ and a deleted neighborhood $W$ of $c$ such that for all $x \in I \cap W$, we have $|f(x)| \le M |g(x)|$.

If $f_1 = O(g)$ and $f_2 = O(g)$ as $x \to c$, then $f_1 + f_2 = O(g).$

If $f_1 = O(g_1)$ and $f_2 = O(g_2)$ as $x \to c$, then $f_1 f_2 = O(g_1 g_2).$

When $x \to c$, if $f$ and $g$ are of the same order, denoted $f = \Theta(g), \quad x \to c,$ it means:

$$\begin{equation*} f = O(g) \text{ and } g = O(f), \end{equation*}$$

that is, there exist $M > 0$ and a deleted neighborhood $W$ of $c$ such that for all $x \in W$,

$$\begin{equation*} \frac{1}{M} |g(x)| \le |f(x)| \le M |g(x)|. \end{equation*}$$

Small o

When $x \to c$, if $f(x) = o(g(x))$, it means:

For every $\varepsilon > 0$, there exists a deleted neighborhood $W$ of $c$ such that for all $x \in W$,

$$\begin{equation*} |f(x)| \le \varepsilon |g(x)|. \end{equation*}$$

If $f_1 = o(g)$ and $f_2 = o(g)$ as $x \to c$, then $f_1 + f_2 = o(g).$

If $f_1 = o(g_1)$ and $f_2 = O(g_2)$ as $x \to c$, then $f_1 f_2 = o(g_1 g_2).$

Asymptotic Equivalence

When $x \to c$, functions $f$ and $g$ are said to be asymptotically equivalent, written as $f \sim g$, if

$$\begin{equation*} f = g + o(g). \end{equation*}$$

That is, for every $\varepsilon > 0$, there exists a deleted neighborhood $W$ of $c$ such that for all $x \in W$,

$$\begin{equation*} |f(x) - g(x)| \le \varepsilon |g(x)|. \end{equation*}$$

Theorems

1. If $x \to c$, and $f + o(f) = G + o(g), \quad G = O(g),$ then $f = G + o(g)$.
2. If $x \to c$, and $f$ is equivalent to $g$, then $g$ is equivalent to $f$.
3. If $x \to c$, and $f$ is equivalent to $g$, and $g$ is equivalent to $h$, then $f$ is equivalent to $h$.