In the definition of limit we say as we hone in on $x=a$ then $f(x)$ hones in on $L$ on the vertical axis.
While the difference between $x$ and $a$ must be nonzero it says nothing of the sort for $f(x)$ and $L$ on the other axis, just that the distance must be less than epsilon.
So in general when does $|f(x)-L| = 0$? Does it only occur in certain situations? Because if we can actually equal $L$ and yet continue to get closer to $L$ by bringing $x$ even closer to $a$ then what does that even mean since we've already "arrived"?
There is no answer "in general", but it generally can occur... It won't in the special case where $f$ is strictly monotonous near $a$.
For exemple with $f(x)=L$ for all $x\in \mathbb{R}$, you will have allways $| f(x)- L|=0$ ; or with $a=0$, $L=0$ and $f(x)=x\sin(\frac1x)$, you will have, near $a$ a infinity of $x$ such that $|f(x)-L|=0$.