In analysis, the concept of a limit at infinity vs. a limit at a real number $r$.
Typically, there is a whole list of definitions of various limits $\lim_{x \rightarrow a} f(x) = b$, depending on whether $a$ and $b$ are ordinary reals or $\pm \infty$. You may have 9 separate definitions of the limit, one for each case. This situation repeats itself any time a limit is used implicitly, for example if an integral converges to a real or to $\pm \infty$, a series converges, and so on.
Everyone knows that these definitions are really the same, but it seems more cumbersome to have a single unified definition than to have separate definitions that are, informally, the same concept. It is this covert "intuitive sense" in which all the definitions are the same that is not clearly defined.
