First of all, in the usual model of the real line there are no infinitesimal numbers (i.e. positive numbers that are smaller than 1/n, for any positive integer n). They don't exist.

Before the calculus was properly formulated infinitesimals were very popular for a very good reason. They make all the proofs a lot easier! For example Newton and Leibeniz used these "ideas". Eventually, after the work of Weierstrass the familiar epsilon-delta arguments of mathematical analysis that we use today were introduced, and infinitesimals were relegated to the dustbin of history. Although, in high school mathematics infinitesimals are still often employed in plausibility arguments (I will not call them proofs).

All that changed in the 1960s when Abraham Robinson invented the idea of nonstandard analysis and hyperreal numbers. The basic idea is that instead of working with the usual set of real numbers R one adjoins some extra elements to R to form *R the hyperreal line. This contains all the usual real numbers but it also contains infinitesimals (and their reciprocals which are infinitely large).

After doing this it is possible to set the clock back and give the same kind of arguments that Newton gave but now in a completely rigorous way. Those high school arguments also become possible to justify.

This is quite a fun idea and I'm a bit surprised that it hasn't caught on as a way to teach analysis. Probably the reason why it hasn't is that although things are nice once you have the hyperreal line, its construction is a bit abstract and nonintuitive and that perhaps scares people off.

See also smallest number greater than 0.