Editing Limit of a function (section)

===Alternative notation===
Many authors<ref>For example, [https://encyclopediaofmath.org/wiki/Limit Limit] at ''[[Encyclopedia of Mathematics]]''</ref> allow for the [[projectively extended real line]] to be used as a way to include infinite values as well as [[Extended real number line|extended real line]]. With this notation, the extended real line is given as {{tmath|\R \cup \{-\infty, +\infty\} }} and the projectively extended real line is {{tmath|\R \cup \{\infty\} }} where a neighborhood of ∞ is a set of the form <math>\{x: |x| > c\}.</math>  The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases.
As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −&infin;, left, central, right, and +&infin;; three bounds: −&infin;, finite, or +&infin;).  There are also noteworthy pitfalls.  For example, when working with the extended real line, <math>x^{-1}</math> does not possess a central limit (which is normal):

<math display=block>\lim_{x \to 0^{+}}{1\over x} = +\infty, \quad \lim_{x \to 0^{-}}{1\over x} = -\infty.</math>

In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit ''does'' exist in that context:

<math display=block>\lim_{x \to 0^{+}}{1\over x} = \lim_{x \to 0^{-}}{1\over x} = \lim_{x \to 0}{1\over x} = \infty.</math>

In fact there are a plethora of conflicting formal systems in use.
In certain applications of [[Numerical analysis|numerical differentiation and integration]], it is, for example, convenient to have [[Negative zero|signed zeroes]].  
A simple reason has to do with the converse of <math>\lim_{x \to 0^{-}}{x^{-1}} = -\infty,</math> namely, it is convenient for <math>\lim_{x \to -\infty}{x^{-1}} = -0</math> to be considered true.
Such zeroes can be seen as an approximation to [[infinitesimal]]s.