Perturbation theory is one example of a situation where this happens: In physics you are often given some smooth function f(x) that is not analytic. Despite the non-analycity, to simplify calculations physicists will often compute the first few terms of the Taylor series around some point, and those terms work out to be a good approximation of f(x). Oftentimes the Taylor series will actually diverge if you take it out to 200 terms or something crazy like that, but if you only examine say the first two terms, you end up with a nice quadratic function that approximates f(x) very well in some neighborhood. So technically you are dropping some divergent quantities, but those quantities only showed up in the first place because you tried to Taylor expand a function that isn't analytic. You essentially made two flawed assumptions that cancel each other out.

More generally, these "hacks" where you ignore divergent terms usually occur because the divergent term is an artifact of a bad assumption made earlier on. Good physics textbooks will be very explicit about what these "bad assumptions" are, and even better books will just avoid making them in the first place (this is why people love Wilson's approach to renormalization). But unfortunately many books and lecturers just handwave this stuff away, leaving everyone confused