I kind of agree and disagree with that at the same time. Yes, doing approximations like that is perfectly fine when you are looking to solve a practical problem and can observe that the solutions you found actually works. When you need guarantees though, I would like a rigorous derivation. Taylor approximation is fine, you can derive error bounds from that, and so on. Obviously, this might get complicated, but what do we have computers for? They should help us with that.