It possible that "were known in general to lead to paradoxes" would be a more historically accurate phrasing than "without firm foundation".

For east to cite examples, there's "The Analyst" (1734, Berkeley). The basic issue was that infinitesimals needed to be 0 at some points in a calculation and non-0 at others. For a general overview, this seems reasonable. Grandi noticed in 1703 that infinite series did not need to give determinate answers; this was widely known in by the 1730's. Reading the texts, it's fairly clear that the mathematicians working in the field were aware of the issues; they would dress up the initial propositions of their calculi in lots of metaphysics, and then hurry to examples to prove their methods.