Epsilon-delta represents a pair of quantifiers (for all ... there exists ...). Challenge-response. The discrete mathematics take ought to be "hey, this is like a game", because if you iterate the quantification you are actually talking about a game-like structure. In other words, the teaching point is something like the transition from "if you want the answer to agree to five decimal places then you have to go far enough towards the limit", to "I know I can always respond to your challenge because I have a strategy!" Proof that a limit exists is the same as showing that there is such a strategy: but NB that the strategy is no more constructive than the proof is.
In other words talking about limits is just discussion of existence proofs for certain kinds of low-level strategies. Roll over Weierstrass!