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?

The objections to infinitesimals were metaphysical. The Volume 14 of the Encyclopædia Britannica 1911 says: 

The name "infinitesimal" has been applied to the calculus because
  most of the leading results were first obtained by means of arguments
  about "infinitely small" quantities; the "infinitely small" or
  "infinitesimal" quantities were vaguely conceived as being neither
  zero nor finite but in some intermediate, nascent or evanescent,
  state. There was no necessity for this confused conception, and it
  came to be understood that it can be dispensed with; but the calculus
  was not developed by its first founders in accordance with logical
  principles from precisely defined notions, and it gained adherents
  rather through the impressiveness and variety of the results that
  could be obtained by using it than through the cogency of the
  arguments by which it was established.

Initially, mathematicians considered negative numbers to be "absurd". Later, complex numbers were considered "absurd" too. In 1911, no mathematician considered negative or complex numbers as absurds. They were already accepted. Surprisingly, infinitesimals were considered to be "absurd"!
How come people could accept things like $\sqrt{-1}$ but could not accept infinitesimals? I think the answer is because of prejudice against them!
Archimedes was the first to use infinitesimals. He considered that they lacked rigor, and then, although he often used them to obtain his results, when he published proofs of these results, he never mentioned infinitesimals! Instead, he used the method of exhaustion.
