http://math.stackexchange.com/questions/1086713/is-0-an-infinitesimal In dictionary , Infinitesimal means ‘an indefinitely small quantity; a value approaching zero’, but natural language is a bad reference for mathematical definitions; it’s 'optimized’ for quickly conveying meaning in 'natural’ settings, not for expressing things precisely. In the context of nonstandard analysis, 0 should certainly be infinitesimal. Even in the limit case, you write “as x→0 f(x)→a” and you intend this to hold in the case that f(x) is constant equal to a. Therefore you need 0 to be considered infinitesimal if you want f(x)→a to be a formalization of infinitesimals. Basically it ends up making more sense to exclude zero in the cases where it should be excluded, then to explicitly include it where it should be included. Stylistically speaking, “nonzero infinitesimal” is much less clunky than “infinitesimal or zero”.
