Strangely, the idea of infinity is one which has been put to incredibly fruitful use in mathematics. (Not just one idea either. There are lots of different meaning which are given to "infinite" throughout mathematics. In this course we'll see quite a few.)

However, infinitesimals haven't worked out so well. When you think in a loose sort of way about the two concepts, they seem somewhat similar (I think the final line of The Incredible Shrinking Man is that the infinitely large and the infinitely small are really the same thing). Try writing down as precise a definition as you can of what you think an infinitesimal number should be. Are there many such numbers, or just one? What about the square of an infinitesimal? Bishop Berkeley wrote a famous attack against natural science, and calculus in particular, called The Analyst . You might like to look it up.

As a final notes, in recent decades, a field called Nonstandard Analysis has been discovered, which does, in fact enable you to talk precisely about infinitesimals, and to do calculus without limits.