Let $f:(0,1) \to \mathbb{R}$ be a given function. Explain how the following definition is not equivalent to the definition of the limit
$\lim\limits_{x \to x_0} f(x) = L$
of $f$ at $x_0 \in [0,1]$ .
For any $\epsilon \gt 0$ there exists $\delta \gt 0$ such that for all $x \in (0,1)$ and $|x-x_0| \leq \delta$, one has $|f(x) - L| < \varepsilon$ .
I don't see the difference between this definition and the actual definition of the limit of a function. Could someone help me out with this?
Please and thank you.