View source for One-sided limit

The [[Limit|limit]] of a function at a point from the right or left. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682501.png" /> be a mapping from an ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682502.png" /> (for example, a set lying in the real line), regarded as a topological space with the topology generated by the order relation, into a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682503.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682504.png" />. The limit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682505.png" /> with respect to any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682506.png" /> is called the limit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682507.png" /> on the left, and is denoted by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682508.png" /></td> </tr></table>

(it does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o0682509.png" />), and the limit with respect to the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825010.png" /> is called the limit on the right, and is denoted by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825011.png" /></td> </tr></table>

(it does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825012.png" />). If the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825013.png" /> is a limit point both on the left and the right for the domain of definition of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825014.png" />, then the usual limit

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825015.png" /></td> </tr></table>

with respect to a deleted neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825016.png" /> (in this case it is also called a two-sided limit, in contrast to the one-sided limits) exists if and only if both of the left and right one-sided limits exist at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825017.png" /> and they are equal.



====Comments====
Instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825018.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825019.png" />) one also finds the notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825021.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068250/o06825023.png" />).