One-sided limit

A one-sided limit for a function $f:D\to\mathbb R$ at $a\in\mathbb R$ is a number $L\in\mathbb R$ satisfying an $\varepsilon$–$\delta$ condition with $x$ restricted to one side of $a$:

• The right-hand limit $\lim_{x\to a^+} f(x)=L$ means: for every $\varepsilon>0$ there exists $\delta>0$ such that if $x\in D$ and $0<x-a<\delta$, then $|f(x)-L|<\varepsilon$.
• The left-hand limit $\lim_{x\to a^-} f(x)=L$ means: for every $\varepsilon>0$ there exists $\delta>0$ such that if $x\in D$ and $0<a-x<\delta$, then $|f(x)-L|<\varepsilon$.

One-sided limits refine the limit at a point by tracking behavior on half-neighborhoods, and they are especially natural for functions defined on intervals with endpoints. They are used to describe jump behavior at a discontinuity point .

Examples:

• For $f(x)=|x|$, both $\lim_{x\to 0^-} f(x)$ and $\lim_{x\to 0^+} f(x)$ equal $0$.
• For $g(x)=\begin{cases}1,&x>0\\-1,&x<0\end{cases}$, one has $\lim_{x\to 0^+} g(x)=1$ and $\lim_{x\to 0^-} g(x)=-1$.