Having never been a mathematics educator, my question could be stupid and, if this is the case, please delete it.
When I was young, from the very beginning of limits, we were teached that there are two things to look at simultaneously : what is the limit and how it is approached. This means that, even when quite young, the concept of asymptotics was in our minds and, thinking back, I have the personal feeling that this was a great idea.
Visiting almost daily MSE, I have the feeling that this is not the case almost anywhere.
So my question : am I totally out of date ? Should I be wrong if I tried to teach that way ?