Limit at a point

A limit at a point for a function $f:D\to\mathbb R$ (with $D\subseteq\mathbb R$) at a point $a\in\mathbb R$ is a number $L\in\mathbb R$ such that: for every $\varepsilon>0$ there exists $\delta>0$ with the property that whenever $x\in D$ and $0<|x-a|<\delta$, one has $|f(x)-L|<\varepsilon$. Typically one assumes that $a$ is a limit point of $D$.

The inequalities use the absolute value to measure distance on the real line. One-sided variants are captured by the one-sided limit .

Examples:

• If $f(x)=x^2$, then $\lim_{x\to 2} f(x)=4$.
• If $g(x)=\begin{cases}1,&x>0\\-1,&x<0\end{cases}$ on $D=\mathbb R\setminus\{0\}$, then $\lim_{x\to 0} g(x)$ does not exist.