This has come up before, but it's one of the real key points so let's take it up again.
Any answer that you write on this (or any other SE) site should be motivated by a sincere desire to be helpful to someone. This does not imply (in my opinion) that your answer has to be appropriate for the OP, because for most questions on this site the OP is just one of a much larger number of mathematics students who will be reading it. It is a matter of judgment when to leave a "higher level" answer; after a certain point I think you have to revisit the idea of having a sincere desire to help. If your answer to what seems to be a standard freshman calculus question involves measures and distributions on infinite-dimensional manifolds, then maybe you should look for a better spot to expose this knowledge: the people who would actually benefit from it may not be the ones who are reading the question at all.
What you are talking about seems slightly different to me: for pedagogical reasons you are leaving an answer which is more intuitive than precise (and in fact, when made precise may not be literally true). I think this is certainly part of giving good answers to a certain level of question, and more generally, part of being a good math teacher: part of the balancing act of knowing advanced mathematics (e.g. having a PhD in the subject) and teaching freshman calculus is that students can benefit from the depths and rigor of your subject knowledge --- but only if you show a lot of restraint, otherwise you end up snowing them altogether.
Where the right balance is depends on essentially everything in sight, so it is hard to say objectively, and well-meaning experts may disagree. I try to never say anything that from a more advanced perspective would turn out to be wrong (thus I am usually either very precise or really vague in what I tell students). Not everyone believes this is a good way to teach. In fact, I sort of believe that I would be a better teacher if I were able to lie a little more.
I remember once attending a talk for graduate students by a fellow young faculty member. He was talking about homotopy theory, and he mentioned that if two spaces had the same homotopy groups they had to be homotopy equivalent. I interrupted to ask whether I had heard him correctly, because at least the isomorphisms on homotopy groups should be induced by a map on the spaces. His response to that was memorable: "Shut up, Pete." I asked him whether I had heard his response correctly. I had [my last question had been in the way of a joke] and the talk moved on happily. This person was hired into a position that took advantage of his especially strong teaching skills, and his research expertise was algebraic topology [and mine is not]. In other words, he made a different pedagogical choice than what I would have made, and both his teaching skills and subject area knowledge were such that I have every reason to suspect that his choice was correct and mine would not have been. (But I still would never say that, and I can be the guy who pipes up on this site to mention that some pedagogically useful answer has a minor technical flaw. Sorry; we are who we are.)
To bring things back to the example: I would never say "infinitely close" in a calculus class, because it sounds like it should mean something, and it could, but I don't mean it to mean what it could in the sense of infinitesimals (or at least, not necessarily). Instead I would give the epsilon-delta definition at least once -- again, pretty much bad teaching that I can't get away from -- do my best to explain it geometrically at least once, and then use very vague language like closer and closer from then on. (Or in response to a persistent question I would volunteer to give the formal definition again, which is pretty much a showstopper.)
By the way, the correct answer to "Is the derivative [or the value of some other kind of limit] only an approximation?" is surely no, right? To say that it is seems to me to be committing a much larger teaching sin: to be overly wedded to an overly formalistic interpretation/understanding of a more advanced mathematical topic. (This is roughly the sort of sin that some calculus texts commit when they try to prove a big theorem: they expose their own knowledge of advanced calculus to be a bit more fragile and formalistic than I would like it to be.) Yes, you can do a nonstandard analysis calculation where at the end you take the standard part of a nonstandard real number, and in that precise technical sense the derivative might actually be an infinitesimal approximation. But then you take the standard part function and get the exact (in the standard sense!) answer. Unless you are actually teaching a nonstandard calculus class in which you are letting the students in to this technicality, you shouldn't hold onto it yourself when you are teaching calculus.
