instead of division with absolute value.
What you're saying is that in any open ball centered at 0, we can find a value of f(x) that is -1, and a value of f(x) that is 1. So we will never be able to get |f(x)-L| bounded for any epsilon less than or equal to 1. (If epsilon is greater than 1, then we can always choose L = 0 and the bound holds.)
Another approach is simply to say the right- and left-hand limits exist and are not equal.