Definition of $\limsup$ from the right or from the left

Let $X$ be a metric space and $E \subset X$. If $f: E \rightarrow \mathbb{R}$, we can define for $a \in E$, the superior limit\begin{align}\limsup_{x \to a} f(x) = \lim_{r \to 0} \left( \sup\{ f(x) : x \in E \cap B(a,r) \setminus \{a\} \} \right)\end{align}here $B(a,r)$ denotes the open ball of radiuos $r$ centered at $a$.

Now my question is what does it mean ''superior limit from the right''. I know that now $X$ must be $\mathbb{R}$ and $E$ is a real subset (probably open). By definition I proposed to myself\begin{align}\limsup_{x \to a^+} f(x) = \lim_{r \to 0^+} \left( \sup\{ f(x) : a < x < a + r \} \right)\end{align}but I don't know if this is right. I suppose that similar definitions can be given for superior limit from the left and for inferior limit from the left and from the right.