real analysis - Why do we need $x_0$ to be a cluster point if we take take the limit $lim_{x to x_0} f(x)$?

We did limits of functions recently and I am wondering why we always required that $x_0$ is a cluster point of the domain. Why would taking the limit not work if $x_0$ is not a cluster point?

Our definition of a limit of a function is

Let $D \subseteq \mathbb R$ be a subset, $x_0$ a cluster point of $D$ and $f: D \to \mathbb R$ a function. We say $f$ converges to $L \in \mathbb R$ and write $\lim_{x \to x_0} f(x) = L$ $\iff \forall \varepsilon \gt 0 \, \exists \delta \gt 0 \, \forall x \in D \setminus \{x_0\}: |x-x_0| \lt \delta \implies |f(x)-L| \lt \varepsilon$

Our definition of a cluster point is

Let $D \subseteq \mathbb R$ be a subset and $x_0\in \mathbb {R}$. We say $x_0$ is a cluster point of $D$ $\iff$ for every $\delta \gt 0$ we have $D \cap (x_0-\delta, x_0-\delta) \setminus \{x_0\} \neq \emptyset$