Players agree on a function f(x) and enter it. For some input value x=c (also entered by the players), a "candidate limit" value L is entered, as the alleged limit of f(x) as x approaches the value c.
Player 1 chooses a "target tolerance" epsilon to define an interval about L. The goal of Player 2 is to choose "response input distance" delta to define an interval around x=c, with the goal that all of the graph between the blue dotted lines defined by delta should lie inside the green "Goal Box".

Player 1 is "giving a number epsilon greater than zero", for which Player 2 strives to produce "a number delta greater than zero" with the following goal:
For all x satisfying
(c - delta) < x < (c + delta)
the output values of the function f will satisfy the inequality
(L - epsilon)< f(x) < (L + epsilon).
To say the L is the limit of f(x) as x approaches c is, in terms of the game, to say that Player 2 can always win, for any tolerance value epsilon that Player 1 happens to set. If Player 1 can set a tolerance value so that Player 2 cannot win, then L is not the limit.