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From the $\epsilon-\delta$ definition of a limit, we can see that any limit can be broken up into two "one-sided limits". These "one-sided" limits are simple cases that arise as a consequence of the absolute value of $|x-a|$ in the formal definition of a limit.

Stated Mathematically :$$\left(\lim_{x \ \to\ a} f(x) = L\right) \Leftrightarrow \left(\lim_{x \ \to \ a^+} f(x) = L\right) \land \left(\lim_{x \ \to \ a^-} f(x) = L\right)$$

Defined Rigorously Using $\epsilon-\delta$ :

$$\left(\forall \epsilon > 0\ (\exists\ \delta > 0 : 0 <|x-a|<\delta \implies |f(x)-L|<\epsilon)\right) \Leftrightarrow \left[(\forall \epsilon > 0\ (\exists\ \delta > 0 : a <x<a +\delta \implies |f(x)-L|<\epsilon) \land \ (\forall \epsilon > 0\ (\exists\ \delta > 0 : a-\delta <x<a \implies |f(x)-L|<\epsilon)\right] $$


Is there better terminology to distinguish between limits?

But is this what the limits are formally called? Are "one-sided" limits always called/referred to as "one-sided" limits (i.e. as the "left-side limit" or "right-side limit") even in higher mathematics? Surely the concept of "one-sidedness" gets thrown out the window once you move past Real Analysis, as the only reason we can make this, ultimately false, visual analogy is due to properties of the Real Number Line?

And what do you call the original limit then, that can be broken up into two "one-sided" limits? Do you just refer to it as the "Limit as $x$ approaches $a$ of $f(x)$", the whole time? Furthermore is a one-sided limit always referred to as : "Limit as $x$ approaches $a$ from the left/right of $f(x)$"

Is the convention in terminology in higher mathematics kept constant, especially when talking about limits, throughout various areas in higher mathematics, for example going from Mathematical Analysis (Real, Complex, Functional etc.) to Topology etc?

I realize that all of this can be easily solved by writing out what is trying to be said Mathematically (as I did above, which is what I prefer to do), but let's assume for the pure terminology purposes of this question that we can't. What is the proper terminology to use in that case to refer to the different types of limits?

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  • $\begingroup$ Very similar to a question you asked only an hour ago (with much of it copy and pasted). $\endgroup$ – Daniel W. Farlow Apr 27 '16 at 13:40
  • $\begingroup$ @Daniel In that question, I was asking about a rigorous Mathematical definition, of one-sided limits. Here I am asking a question purely on terminology, without any notation involved They are similar, but not the same. The only part that was copy-pasted was the LaTeX code for the opening paragraph of the post. $\endgroup$ – Perturbative Apr 27 '16 at 13:47
  • $\begingroup$ In higher mathematics you consider limits in spaces different than the line... In such spaces, the concept of "left/right sided" limit makes no sense. $\endgroup$ – Gabriel Romon Apr 27 '16 at 14:06

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