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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

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

(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}  } $).