Work from the formal definition of a limit. The formal definition of a limit is: The limit of f(x) is L as x approaches k if for every epsilon greater than zero there is a corresponding delta greater than zero such that when the absolute value of the difference x and k is less than delta it makes the absolute value of the difference between f(x) and L less than epsilon. More informally, this means that the limit of f(x) is L as x approaches k if you can make f(x) as close to L as you want by making x close enough to k. To do epsilon delta proofs, you must show that you can define delta in terms of epsilon for a given function and limit.
Manipulate "|f(x) - L| is less then epsilon" until you get "|x - k| is less than something." Let this "something" be delta and you have the relationship. Keep in mind the formal definition and the central idea, which is that you want to show that for any epsilon there is a delta that makes the definition work; you must define delta in terms of epsilon.
Look at several examples to get a feeling for how these proofs proceed. For example, to prove that the limit, as x approaches 1, of 3x - 1 is 2, we let k = 1, L = 2 and f(x) = 3x - 1. To make sure |f(x) - L| is less than epsilon, make |(3x - 1) - 2| less than epsilon. Which means that |3x - 3| is less than epsilon, and 3 |x - 1| is less than epsilon or ||x - 1| is less than epsilon/3. So if we let delta = epsilon/3, |f(x) - L| is less than epsilon whenever |x - k| is less than delta.