One-sided limit

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, $\lim\limits _ {x \rightarrow x _ {0} - }$, $\lim\limits _ {x \uparrow x _ {0} }$).