The definition in my book is as follows:
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 $0<|x-c|<\delta$, ...
I have just edited it, I think it is better than the original title "$\epsilon - \delta$ definition of a limit?" But if you have some ideas about title which describes the question better, go ahead.
The definition in my book is as follows:
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 $0<|x-c|<\delta$, ...
