Illustration of epsilon-delta definition of the limit.
Recall that when we say the limit as approaches of is , written , we mean that for every there is a with
whenever .
This definition is often hard to understand the first time you see it, so it is useful to do a visual translation and think of it like a game -
Pick any value for and set up a target (interval) around of radius . Now see if you can find an interval around , say of radius so the interval looks like , for which any value of in this interval must get sent to the -interval around .
If, for every you choose you can find such a , then .
To play with the following applet, try setting to be some number, preferably small. Then see if you can find a value for by dragging the slider so that each value that starts in the blue -interval gets sent to the red -interval. Drag the purple slider to test mapping different values and see where they end up.