My teacher gave this as the definition of a limit:
$$\lim_{x\to c} f(x) =L$$ if for each $\epsilon >0$, there is some $\delta >0$ such that $|f(x)-L|<\epsilon$ as long as $x \in (c-\delta, c+\delta ).$
Does the phrase "as long as" work here? In other texts I see $$\forall x, |x-c|< \delta \implies |f(x)-L|<\epsilon$$ and sometimes $|f(x)-L|<\epsilon$ whenever $x \in (c-\delta, c+\delta ).$ I guess my question is twofold: Is my teacher correct in saying "as long as" here? Also, did he need to say $x\in (c-\delta, c+\delta )$ for each $x$
"Whenever" is a clearer phrasing to use. The intention of "as long as" is that $|f(x) - L| < \epsilon$ for any value of $x$ satisfying $x \in (c-\delta, c+\delta)$, but it is not quite as clear.
As for the definition's use of the interval notation instead of $|x - c| < \delta$, that is inconsequential. They are equivalent.
However, the more glaring omission is that the $\delta$-neighborhood $(c-\delta, c+\delta)$ needs to be punctured: we must write $$x \in (c - \delta, c + \delta) \setminus c,$$ or $$0 < |x - c| < \delta.$$ You cannot include $x = c$ because $f$ is not necessarily defined at $c$.