The informal 17th century arguments involving infintesimals and infinities
-can- be made precise, but they require some logic to make sense.
Or rather, you need some model theory in order to push the theorems down to
normal real numbers, and in order to construct infintesimals.

EG, for all $n$, there exists $\epsilon$ such that $\epsilon < \frac{1}{n}$;
hence by set theory proof, there is a model which has an $\epsilon < \frac{1}{n}$
for all $n$.

However, it's very satisfying to be able to reason (rigorous!) like Leibniz:
$(x+\dx)^2=x^2+2x\dx+(\dx)^2$, so
$\frac{(x+\dx)^2 - x^2}{\dx}=2x+\dx \cong 2x$.
(i.e., don't need to take limits, quite).
(of course in order to make sense of non-standard analysis, you need some
notion of limit, but it's not a very sophisticated notion you need: just going
to zero)

This answers another question I had, which is the analogy between -growth- of
functions and -ordinals-:
an infinity in non-standard analysis is a sequence, growing at a certain rate;
ordinals are hence equivalence classes of these
Hence,
$\omega = (1,2,3,\dotsc)$,
and we can also do $\omega+1 = (2,3,4,\dotsc)$
and $\omega-1$ (which you \emph{can't} do with logic cardinals);
similarly $\omega^2$ makes sense, and $2^\omega$, and so forth.
(also get $\epsilon = (\frac{1}{1},\frac{1}{2},\frac{1}{3},\dots) = 1/\omega$)

Yay!! You can do arithmetic with ordinals in a good way...
