In Wikipedia, the definition of limit superior and limit inferior of functions from metric spaces is:
Take a metric space $X$, a subspace $E$ contained in $X$, and a function $f:E\to \mathbb {R}$. Define, for any limit point $a$ of $E$, $$\limsup _{x\to a}f(x)=\lim _{\varepsilon \to 0}\left(\sup \,\{f(x):x\in E\cap B(a,\varepsilon )\setminus \{a\}\}\right)$$ and $$\liminf _{x\to a}f(x)=\lim _{\varepsilon \to 0}\left(\inf \,\{f(x):x\in E\cap B(a,\varepsilon )\setminus \{a\}\}\right)$$ where $$B(a,\varepsilon )$$ denotes the metric ball of radius $\varepsilon$ about $a$.
From the definition above, why do we need the point $a$ is a limit point of $E$?
Thanks for any explanation.