The most salient to me would be Cauchy's 1821 "proof" that the pointwise limit of continuous functions is continuous; counterexamples were not constructed until 1826 (by which time functions were better understood) and it took until 1853 for the actual conditions (uniform continuity) to be developed properly. This statement was at least as well supported in 1821 as Euler's was in 1735.
What was the evidence?
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Could you give a source for this claim? "Foundation" sounds to me anachronistic for 1735.
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.