Are my definitions of limits accurate?

Question

I wrote the definitions below. Are they accurate? If not, what correction(s) should be made? You may think based on an apparent lower bound for my level of mathematical maturity that I could answer this question for myself, but peer review is important to me.

The $\varepsilon$-$\delta$ definition of a limit. Let $f$ be a function defined on an open interval containing a number $a$ except possibly at $a$. Then, for some number $L$, \begin{equation*} \lim_{x \rightarrow a} f\left(x\right) = L \end{equation*} if $\forall \varepsilon > 0, \exists \delta > 0$ such that $\forall x$ in the domain of $f$, \begin{equation*} |x - a| < \delta \implies |f\left(x\right) - L| < \varepsilon. \end{equation*} The $\varepsilon$-$\delta$ definition of a left-hand limit. Let $f$ be a function defined on an interval with a number $a$ as its right endpoint. Then, for some number $L$, \begin{equation*} \lim_{x \rightarrow a^-} f\left(x\right) = L \end{equation*} if $\forall \varepsilon > 0, \exists \delta > 0$ such that $\forall x$ in the domain of $f$, \begin{equation*} a - \delta < x < a \implies |f\left(x\right) - L| < \varepsilon. \end{equation*}

Answer 1

It is almost totally correct. The $\varepsilon-\delta$ part is fine. The only problem is that the definition of left-hand limit doesn't requere that $a$ is the right endpoint of the domain of $f$.

Answer 2

For the first one, I think it's more common to use the definition with $0 < |x-a| < \delta$ instead of $|x-a|<\delta$, although you can find both variants in different sources. In other words, even if $f(a)$ happens to be defined, that value shouldn't be taken into account as far as the limit is concerned. (With your definition, the limit must equal $f(a)$ if $f(a)$ is defined.)

Also, requiring $f$ to be defined on a punctured interval around $a$ is a bit strong; it's enough that $a$ is a limit point of the domain of $f$.