Infinitely-small function

A function of a variable $x$ whose absolute value becomes and remains smaller than any given number as a result of variation of $x$. More exactly, a function $f$ defined in a neighbourhood of a point $x_0$ is called an infinitely-small function as $x$ tends to $x_0$ if for any number $\varepsilon>0$ it is possible to find a number $\delta=\delta(\varepsilon)>0$ such that $|f(x)|<\varepsilon$ is true for all $x$ satisfying the condition $|x-x_0|<\delta$. This fact can be written as follows: $$\begin{equation} \lim_{x\to x_0}f(x)=0. \end{equation}$$ Further, the symbolic notation $$\begin{equation} \lim_{x\to+\infty}f(x)=0 \end{equation}$$ means that for any $\varepsilon>0$ it is possible to find an $N=N(\varepsilon)>0$ such that for all $x>N$ the inequality $|f(x)|<\varepsilon$ is true. The concept of an infinitely-small function may serve as a base of the general definition of the limit of a function. In fact, the limit of the function $f$ as $x\to x_0$ is finite and equal to $A$ if and only if $$\begin{equation} \lim_{x\to x_0} f(x)-A=0, \end{equation}$$ i.e. if the function $f(x)-A$ is infinitely small. See also Infinitesimal calculus.