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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.