Editing Extended real number line (section)

==Miscellaneous==

Several functions can be [[continuity (topology)|continuously]] [[restriction (mathematics)|extended]] to <math>\overline\R</math> by taking limits. For instance, one may define the extremal points of the following functions as:
:<math>\exp(-\infty)=0</math>,
:<math>\ln(0)=-\infty</math>,
:<math>\tanh(\pm\infty)=\pm1</math>,
:<math>\arctan(\pm\infty)= \pm\frac{\pi}{2}</math>.

Some [[singularity (mathematics)|singularities]] may additionally be removed. For example, the function <math>1/x^2</math> can be continuously extended to <math>\overline\R</math> (under ''some'' definitions of continuity), by setting the value to <math>+\infty</math> for <math>x=0</math>, and 0 for <math>x=+\infty</math> and <math>x=-\infty</math>. On the other hand, the function <math>1/x</math> can''not'' be continuously extended, because the function approaches <math>-\infty</math> as <math>x</math> approaches 0 [[one-sided limit|from below]], and <math>+\infty</math> as <math>x</math> approaches 0 from above, i.e., the function not converging to the same value as its independent variable approaching to the same domain element from both the positive and negative value sides.

A similar but different real-line system, the [[projectively extended real line]], does not distinguish between <math>+\infty</math> and <math>-\infty</math> (i.e. infinity is unsigned).<ref name=":2">{{Cite web|url=http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html|title=Projectively Extended Real Numbers|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref> As a result, a function may have limit <math>\infty</math> on the projectively extended real line, while in the extended real number system only the [[absolute value]] of the function has a limit, e.g. in the case of the function <math>1/x</math> at <math>x=0</math>. On the other hand, on the projectively extended real line, <math>\lim_{x\to-\infty}{f(x)}</math> and <math>\lim_{x\to+\infty}{f(x)}</math> correspond to only a limit from the right and one from the left, respectively, with the full limit only existing when the two are equal. Thus, the functions <math>e^x</math> and <math>\arctan(x)</math> cannot be made continuous at <math>x=\infty</math> on the projectively extended real line.