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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
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<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>
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(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
+
The [[Limit|limit]] of a function at a point from the right or left. Let  $  f $
 +
be a mapping from an ordered set  $  X $(
 +
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  $  Y $,
 +
and let  $  x _ {0} \in X $.
 +
The limit of  $  f $
 +
with respect to any interval $  ( a, x _ {0} ) = \{ {x } : {x \in X,  a < x < x _ {0} } \} $
 +
is called the limit of  $  f $
 +
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/o06825011.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow x _ {0} - 0 }  f ( x)
 +
$$
  
(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
+
(it does not depend on the choice of $  a < x _ {0} $),
 +
and the limit with respect to the interval  $  ( x _ {0} , b) = \{ {x } : {x \in X,  x _ {0} < x < b } \} $
 +
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/o06825015.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow x _ {0} + 0 }  f ( x)
 +
$$
  
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.
+
(it does not depend on the choice of $  b > x _ {0} $).  
 +
If the point  $  x _ {0} $
 +
is a limit point both on the left and the right for the domain of definition of the function  $  f $,
 +
then the usual limit
  
 +
$$
 +
\lim\limits _ {x \rightarrow x _ {0} }  f ( x)
 +
$$
  
 +
with respect to a deleted neighbourhood of  $  x _ {0} $(
 +
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  $  x _ {0} $
 +
and they are equal.
  
 
====Comments====
 
====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" />).
+
Instead of $  \lim\limits _ {x \rightarrow x _ {0}  + 0 } $(
 +
respectively, $  \lim\limits _ {x \rightarrow x _ {0}  - 0 } $)  
 +
one also finds the notations $  \lim\limits _ {x \rightarrow x _ {0}  + } $,  
 +
$  \lim\limits _ {x \downarrow x _ {0}  } $(
 +
respectively, $  \lim\limits _ {x \rightarrow x _ {0}  - } $,  
 +
$  \lim\limits _ {x \uparrow x _ {0}  } $).

Latest revision as of 08:04, 6 June 2020


The limit of a function at a point from the right or left. Let $ f $ be a mapping from an ordered set $ X $( 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 $ Y $, and let $ x _ {0} \in X $. The limit of $ f $ with respect to any interval $ ( a, x _ {0} ) = \{ {x } : {x \in X, a < x < x _ {0} } \} $ is called the limit of $ f $ on the left, and is denoted by

$$ \lim\limits _ {x \rightarrow x _ {0} - 0 } f ( x) $$

(it does not depend on the choice of $ a < x _ {0} $), and the limit with respect to the interval $ ( x _ {0} , b) = \{ {x } : {x \in X, x _ {0} < x < b } \} $ is called the limit on the right, and is denoted by

$$ \lim\limits _ {x \rightarrow x _ {0} + 0 } f ( x) $$

(it does not depend on the choice of $ b > x _ {0} $). If the point $ x _ {0} $ is a limit point both on the left and the right for the domain of definition of the function $ f $, then the usual limit

$$ \lim\limits _ {x \rightarrow x _ {0} } f ( x) $$

with respect to a deleted neighbourhood of $ x _ {0} $( 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 $ x _ {0} $ and they are equal.

Comments

Instead of $ \lim\limits _ {x \rightarrow x _ {0} + 0 } $( respectively, $ \lim\limits _ {x \rightarrow x _ {0} - 0 } $) one also finds the notations $ \lim\limits _ {x \rightarrow x _ {0} + } $, $ \lim\limits _ {x \downarrow x _ {0} } $( respectively, $ \lim\limits _ {x \rightarrow x _ {0} - } $, $ \lim\limits _ {x \uparrow x _ {0} } $).

How to Cite This Entry:
One-sided limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-sided_limit&oldid=48045
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article