I'm not sure what flaws the other answer is finding with your idea. This is a perfectly interesting question. For better or worse, the real answer is a mathematical object is whatever type you can make of it. One thing can be a different type in different contexts, so it is quite useful to ask, in general, "Can I write this another way?"

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You do want to be careful with wording though: a "limit" is just a value, whatever value you get. But you're asking about *taking a limit*. There's no reason you can't conceive of that as a function. (Typically a map that takes functions as input and gives numbers as output are called **functionals**, and Functional Analysis is a major field of math.) There is also no problem with mapping things to infinity. (That's evidenced by limits doing just that! Check out the extended reals on Wikipedia.)

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Of course, the implicit question you may be asking is whether this conception is useful. Not off the top of my head, but people have found uses for the functional F_{p}(f) := f(p), which takes a function and gives its value. So maybe you're on to something! The general philosophy is "if you want to understand something, hit it with a function". So perhaps the question is not whether something *is* a function, but what other functions can tell you. Moreover, if thinking of limits in this way helps you because that's the framework you have already built for yourself, then it is useful. Elegance is overrated in the math community.