Definition of a limit

Let $f$ be a function defined on an open interval containing $c$ (except possibly at $c$ ) and let $L$ be a real number. The statement

$$\lim_{x \to c}f(x)=L$$

means that for each $\epsilon >0$ there exists a $\delta >0$ such that if

$$0<\left\lvert x-c \right\rvert <\delta, \quad\left\lvert f(x)-L \right\rvert <\epsilon.$$