What problems arise when using infinitesimals in calculus? In contemporary real analysis we use a limit definition in terms of deltas and epsilons.
Before that, people used infinitesimals to calculate limits.
Is there a specific non-philosophical reason why we didn't keep on using infinitesimals?
I.e. are there concrete examples in which the use of infinitesimals lead to serious problems?

This is intended as both an answer and a comment. @Conifold, The idea that "the way Leibniz and others were manipulating infinitesimals to find derivatives, for example, led to contradictions" is a misconception. There is a number of recent articles on this in journals ranging from Notices AMS to Erkenntnis. The articles are a bit long but if you are looking for a concise summary you could consult this review by Marcel Guillaume.
