Limits at infinity seem like the right place to start ...
I found almost exactly that satisfaction of unity of the various kinds of limits in the treatment by McShane and Botts in Real Analysis. He uses the appealing term ``direction'' that appears to me to be what Bourbaki calls a filter base. He speaks of converging in direction D where in the smooth function case, D is typically the neighborhoods of a point and in the sequence case, it is the numbers greater than n for an ascending sequence of ns. The book is in the Van Nostrand undergraduate series but I don't know what year it was meant for; I would suspect junior. Certainly I read it after I had read more traditional treatments like Hardy. Whit