adj. 1a. (usually of an increment) approaching zero as a limit, arbitrarily small. Informally, infinitely small.
b. (as substantive) an infinitesimal increment or quantity. In early treatments of the calculus, a derivative was treated as a ratio of infinitesimals and an integral as a sum of products of infinitesimals. These had to be non-zero for the ratio to be well-defined and the products to be non-zero, but zero for the rate of change so derived to be instantaneous and the upper and lower sums to be equal. This paradoxical conception was later largely abandoned in favour of the epsilon-delta treatment of limits, but see hyper-real numbers. 2. in non-standard analysis, having zero standard part. See Archimedean property.