Hi. I was following the OP alright until you came to this: *** |f(x) - L| < *** Can you explain the notation and the ideas?
I hope I can clarify it correctly: it mean that for any arbitrary positive number around L (interpret as a distance around L, and L is the image of a function f for a given input, which is a point of the y axis), there has to be a M > 0, such that if x > M, then the image of f(x) is inside the range of around L.