This worksheet illustrates the formal ([math]\epsilon{-}\delta[/math]) definition of the limit of a real-valued function.

[list=1] [*] Move the point [color=#0000ff][math]a[/math][/color] and adjust [math]\epsilon[/math]. [*] The [color=#ff0000]red[/color] line on the [math]y[/math]-axis denotes the minimum and maximum values attained by [math]f[/math] over the interval [math](a-\delta,a+\delta)[/math]. [*] Vary [math]\delta[/math] and observe that no matter how small [math]\epsilon[/math], you can always set [math]\delta[/math] so that [math]|f(x)-l|<\epsilon[/math]. That is, the red line can always be shrunk so that it fits between the upper and lower green dots. [/list]