Hey guys. Sorry if i dont make sense because i'm just trying to make sense of this concept !

Ok, so I understand that to prove that a limit exists that there HAS to be a delta value (S) where the distance between x and c is less than S which satisfies that distance between f(x) and L is less than E.

So basically if i can find a delta that is greater than 0 and that if i plug in x+delta, the distance f(x) and L is less than epsilon (sorry if im repeating but im still not sure...)

Also, I understand the graphical approach. That is, if i pick any epsilon value , i should be able to get a distance between delta value (S) and x that is within that distance between f(x) and L.

Anyway, i THINK i understand the conceptual part, but im confused with the actual proof/scratch work, and how it's proving what i said above^

Example:

How does the proof below show that? How does solving for abs( f(x) - L ) = E tell us that this is true. I don't understand any of this, and I've been trying to understand this for hours and watching videos, but it just gets soo confusing.

It doesn't make sense to me at all and i feel hopeless. I need to go to bed now, but I've been trying to really understand this, and it's not going too well. (I already looked at the epsilon-delta notes from the calc forum but still confused..)