Because $|f(x)-L|$ is positive (or zero) by definition. Zero is a permitted value as in the constant function, for example.
In the first inequality, zero is not permitted because we are looked at where the function is headed, not the actual value of the function at the given point.
The statement that $\mathop {\lim }\limits_{x \to a} f(x) = L$ means that for $\varepsilon >0$ $\exists \delta>0$ so that if $0<|x-a|<\delta$ then $|f(x)--L|<\varepsilon~.$
Have you ever asked yourself why $0<|x-a|<\delta~?$ That is to say WHY $\bf{\large 0<|x-a|~?}$
Well, the limit as $x\to a$ means that the number $x$ is close to $a$ BUT NOT EQUAL TO $a$ or $0<|x-a|<\delta$
That is the question you should have been asking!
You should also note that in the question about the continuity of $f$ at $x=a$ we use $|x-a|<\delta$ . WHY?