Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 adjective Immeasurably or incalculably minute.
 adjective Mathematics Capable of having values approaching zero as a limit.
 noun An immeasurably or incalculably minute amount or quantity.
 noun Mathematics A function or variable continuously approaching zero as a limit.
from The Century Dictionary.
 Infinitely or indefinitely small; less than any assignable quantity.
 noun In mathematics, a fictitious quantity so small that by successive additions to itself no sensible quantity, such as the unit of quantity, could ever be generated.
from the GNU version of the Collaborative International Dictionary of English.
 adjective Infinitely or indefinitely small; less than any assignable quantity or value; very small.
 adjective the different and the integral calculus, when developed according to the method used by Leibnitz, who regarded the increments given to variables as infinitesimal.
 noun (Math.) An infinitely small quantity; that which is less than any assignable quantity.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective
Incalculably ,exceedingly , orimmeasurably minute ;vanishingly small.  adjective mathematics Of or pertaining to
values that approachzero as alimit .  adjective informal Very small.
 noun mathematics A
nonzero quantity whosemagnitude is smaller than anypositive number (by definition it is not areal number ).
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 noun (mathematics) a variable that has zero as its limit
 adjective infinitely or immeasurably small
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
Examples

That is the one and only actual meaning of the use of the term infinitesimal by Leibniz.

BTW, "infinitesimal" is an actual word and not made up by Obama.

Twice an infinitesimal is an infinitesimal that is twice as big.

To Jack from Texas, I know the Texas education system must be in terrible shape with Republicans running that state for so long, but infinitesimal is not a made up word, just do a search, here is the definition:

Say, rather, "infinitesimal"  much like the snowflake that starts an avalanche.

This has been achieved in two essentially different ways, the one providing a rigorous formulation of the idea of infinitesimal number, the other of infinitesimal magnitude.

Bolzano also formulated a definition of the derivative of a function free of the notion of infinitesimal (see Bolzano [1950]).

Only by reducing this element of free will to the infinitesimal, that is, by regarding it as an infinitely small quantity, can we convince ourselves of the absolute inaccessibility of the causes, and then instead of seeking causes, history will take the discovery of laws as its problem.

In spite of the glowing reports issued annually from various foreign hospitals for natives, and the undeniable good, though desultory and practically infinitesimal, that is being worked by these institutions, we cannot blind ourselves to the fact that western medical science is not making more rapid strides than many other innovations in the great struggle against Chinese prejudice and distrust.

In particular, that potent instrument called the infinitesimal calculus, which Newton had invented for the investigation of nature, had become so far perfected that Laplace, when he attempted to unravel the movements of the heavenly bodies, found himself provided with a calculus far more efficient than that which had been available to Newton.
xstala commented on the word infinitesimal
Also: Incredibly small; immeasurably or incalculably minute.
December 31, 2007