Illustration of epsilon-delta definition of the limit.[br][br]Recall that when we say the limit as [math]x[/math] approaches [math]a[/math] of [math]f(x)[/math] is [math]L[/math], written [math]\lim_{x \to a} f(x) = L[/math], we mean that for every [math]\epsilon > 0[/math] there is a [math]\delta > 0[/math] with[br][math]|f(x) - L| < \epsilon[/math] whenever [math]0 < | x-a | < \delta[/math].[br][br]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 -[br]Pick any value for [math]\epsilon[/math] and set up a target (interval) around [math]L[/math] of radius [math]\epsilon[/math]. Now see if you can find an interval around [math]a[/math], say of radius [math]\delta[/math] so the interval looks like [math](a-\delta,a+\delta)[/math], for which any value of [math]x[/math] in this interval must get sent to the [math]\epsilon[/math]-interval around [math]L[/math]. [br][br]If, for every [math]\epsilon[/math] you choose you can find such a [math]\delta[/math], then [math]\lim_{x \to a} f(x) = L[/math].[br][br]To play with the following applet, try setting [math]\epsilon[/math] to be some number, preferably small. Then see if you can find a value for [math]\delta[/math] by dragging the slider so that each [math]x[/math] value that starts in the blue [math]\delta[/math]-interval gets sent to the red [math]\epsilon[/math]-interval. Drag the purple slider to test mapping different values and see where they end up.