TITLE: Is $0$ an Infinitesimal?
QUESTION [19 upvotes]: For the definition of Infinitesimal,
wikipedia says

In common speech, an infinitesimal object is an object which is
  smaller than any feasible measurement, but not zero in size; or, so
  small that it cannot be distinguished from zero by any available
  means.

MathWorld says

An infinitesimal is some quantity that is explicitly nonzero and yet
  smaller in absolute value than any real quantity.

BUT I met some definition of Infinitesimal in textbooks says 

If $\lim_{{{x}\to{x}_{{0}}}} f{{\left({x}\right)}}={0}$, then we call
  $f(x)$ is an infinitesimal when ${x}\to{x}_{{0}}$.

I found the textbooks's definition conflict with the above two definitions..
Obviously, $f(x)=0$ satisfy the textbooks's definition ,then can we call 0 an Infinitesimal ?

REPLY [11 votes]: The only context I can think of where an infinitesimal makes sense is in a formalization of nonstandard analysis.  In this context,

$\epsilon \in {}^*\mathbb{R}$ is called infinitesimal if $|\epsilon| < r$ for all positive real numbers $r$.

The set of infinitesimals $I$ is an ideal in the ring of finite hyperreal numbers.  Modding out by the ideal $I$ one gets the original set of real numbers $\mathbb{R}$.  In fact, the important standard part function is the natural ring homomorphism from the finite hyperreals to the reals induced by taking this quotient.
All ideals contain $0$, so in this context one always wants $0$ to be infinitesimal.  Also, most definitions involving infinitesimals (e.g., $f(x) - f(a)$ is infinitesimal whenever $x - a$ is infinitesimal) require that $0$ is infinitesimal.  In cases where we don't want to include $0$, we should explicitly ask for a nonzero infinitesimal.