(ε,δ)-definition of the limit

The (ε,δ)-definition is one of the most well-known definitions of the limit, often taught in low-level calculus classes.

Definition

Let $fx$ be defined for all $x\ne a$ over an open interval containing $a$, and let $L$ be a real number. Then $$\underset{x\to a}{\lim }fx=L$$ if, for every $\epsilon >0$, there exists a $\delta >0$ such that, if $0<|x-a|<\delta$, then $|fx-L|<\epsilon$.

Symbolically, $$\begin{array}{c}\underset{x\to a}{\lim }fx=L\\ ⟺\\ \forall \epsilon >0,\exists \delta >0,\forall x,0<|x-a|<\delta ⟹|fx-L|<\epsilon .\end{array}$$

Intuition

In English, one could interpret the statement "the limit at $a$ equals $L$" saying "give me an an upper bound ($\epsilon$) on the distance between the 'outputs' $fx$ and $L$, and I'll give you an upper bound ($\delta$) on distance between the 'inputs' $x$ and $a$."