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Sounds like you're interested in the projectively extended real numbers, you might also be interested in the affinely extended version too. The projectively extended real numbers are homemorphic (and therefore basically equivalent to, for a lot of purposes) the unit circle, and similarly the affinely extended real numbers are homemorphic to the unit interval. Functions to either of those can't diverge, but they can still oscillate which is another way of not converging. E. g. sin(x) doesn't converge to any extended real at infinity and sin(1/x) cant be extended to a contniuous function into extended reals near zero
