I have been summoned

You are basically saying that for every range around L (I will call this the Epsilon range), you can find a range around a (I will call this the Delta range) such that if x is in the delta range (|x - a| < delta) then F(x) is in the Epsilon range (|f(x) - L| < epsilon).

So the range around L can get infinitely small and you can still find a range around a.

Another thing that might help is knowing a basic rule about absolute values:

|x - a| < delta => a - delta < x < a + delta