Most mathematicians throw a conniption fit if you happen to mention the unfortunate term "infinitesimal".
This actually happened to me in my senior year of high school, when the head of the math department was a substitute teacher in Calc class one day. The merest mention of the word "infinitesimal" and we were subjected to another month of limit theory.
The inventors of calculus imagined differentials as infinitesimals, and used them in their work.
Unforunately, the notion of a quantity that is infinitely small causes some pretty severe paradoxes in arithmetic.
Because of the problems associated with infinitesimals, calculus was reformulated by 18th century mathematicians
to be based upon the theory of limits.
The father of modern set theory, Georg Cantor, called infinitesimals "the Cholera-bacillus of mathematics." This may have due to the fact that the existence of infinitesimals would render his Continuum Hypothesis false.