Just curious, the definition of a limit is:
For every $\epsilon\gt0$, there exists a $\delta\gt0$, such that for every $x$, the expression $0\lt|x-c|\lt\delta$ implies $|f(x)-L|\lt\epsilon$.
Is there a reason why this definition uses an open ball around L and an open, punctured disc around c instead of closed discs/balls? To be more precise, would it be incorrect to say:
For every $\epsilon\gt0$, there exists a $\delta\gt0$, such that for every $x$, the expression $0\lt|x-c|\le\delta$ implies $|f(x)-L|\le\epsilon$.
as the definition of a limit?