For a given function f(x), in order to say the limit of f(x) as x approaches a is L, we need to know that for Every ε > 0, there exists a corresponding δ > 0, such that if |x - a| < δ, then |f(x) - L| < ε.
Use the sliders to change the value of δ to find a value that works with ε=2. Then try shrinking ε and finding a corresponding δ for the new ε.

You can drag the point a around to see behavior at other points.
Hold CNTR and use the mouse and school wheel to change the displayed window.
Initially, f(x) = 1/8 x^3.
Type in "f(x) = " to try a different function.
(Note: This demo assumes f(x) is continuous and monotone in the interval (x - δ, x + δ). )