When dealing with functions, there is usually both a point notation and a point-free notation. For instance:

  • Arithmetic operations on functions: $f(x) + g(x)$ vs $f+g$
  • Function composition: $f(g(x))$ vs $f\circ g$
  • Derivative: $f'$ vs $\frac{d}{dx} f(x)$
  • Integration: $\int_{[a,b]} f$ vs $\int_a^b f(x)\, dx$

However, with the limit, except in one place I have always seen $\lim_{x\to a} f(x)$ or $\lim_{x\to a;\, x\in E} f(x)$ instead of (say) $\lim_{a} f$ or $\lim_{a;\, E} f$. The exception is Spivak's book Calculus, which says (after noting that the limiting variable is a dummy variable):

A more logical symbol would be something like $\displaystyle \lim_a f$, but this notation, despite its brevity, is so infuriatingly rigid that almost no one has seriously tried to use it.

The book then goes on to note that point notation allows one to work with anonymous functions and also to work with multiple variables (e.g. $\lim_{x\to a} (x+t^3)$ vs $\lim_{t \to a} (x+t^3)$). However, both of these reasons apply to point vs point-free notation in general, not just in the case of limits.

Is there some deep (e.g. psychological) reason for limits being an exception in not having a widely-used point-free notation?

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    Just a note: in general, $f(x)+g(x)$ and $f+g$ are used to denote very different things; one a value, one a function. – Sambo Dec 1 at 4:33
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    I suppose part of it is due to the idea of "approaching" things in the set you're taking the limit. So you would take different $x$s in your set, and let the "x"s get closer and closer to $a$, which is especially the way most people are taught precalculus. I suppose this makes it more intuitive. – YiFan Dec 1 at 5:42
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    The fact you often need to specify the direction you're approaching the limit from is probably relevant too. I think that quite often when dealing with limits you're doing something where you want the notation to keep track of as much information as possible. – timtfj Dec 1 at 14:48

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