The familiar definition of a limit is that we say $\displaystyle\lim_{x \rightarrow a} f(x) = L$ when
$$\forall \varepsilon > 0 : \exists \delta > 0 : 0<|x-a|<\delta \Rightarrow |f(x) - L| < \varepsilon.$$
I've been doing a little bit of work with limits lately and have come across a situation where it has been useful to define a new sort of limit, say ${\lim}^\dagger$, that is defined so that we can say $\displaystyle{\lim_{x \rightarrow a}}^\dagger f(x) = L$ when
$$\forall \varepsilon > 0 : \exists \delta > 0 : 0<|x-a|<\delta \Rightarrow \color{red}{0<} |f(x) - L| < \varepsilon.$$
(The modified portion is indicated in red.)
Question: Does this type of stricter limit ${\lim}^\dagger$ have any sort of standard name, notation, or known properties?
Background:
I was doing a bit of toying around with situations when
$$\displaystyle \lim_{x\rightarrow a} g(x) = G \text{ and } \displaystyle \lim_{x \rightarrow G} f(x) = L \text{ imply that } \displaystyle \lim_{x\rightarrow a} f(g(x)) = L. \tag{1}$$
I was able to determine that this seems to hold when $\displaystyle {\lim_{x\rightarrow a}}^\dagger g(x) = G$, with ${\lim}^\dagger$ being as defined above. I don't think this is the only circumstance when (1) holds, but it was at least a straightforward enough starting place in my investigation. So, this is what led me to be asking about ${\lim}^\dagger$.