This project was supported by NSF grant number DUE-9950473.
Understanding Limits
This applet is designed to help you understand the formal epsilon/delta definition of a limit.
The question is whether the limit as x approaches a of a function, f(x), is to L.
Testing this involves finding a delta (greater than zero) for each epsilon (greater than zero)
such that if 0 < |x - a| < delta, then |f(x) - L| < epsilon.
In this applet, the values of epsilon and delta can be adjusted using
sliders or text input boxes at the bottom of the applet.
A horizontal band about y = L is shown in pink and yellow.
This band contains all points (x,y) such that |y - L| < epsilon.
A vertical band about x = a is shown in light green and yellow. This band contains all
points (x,y) such that |x - a| < delta. Note that the yellow rectangle is the
intersection of the delta and epsilon bands. If the limits is L, then for every point
(x,f(x)) in the delta band (except possibly for x = a) must also lie in the epsilon band.
Note: An easy way of saying this is that the graph of the function can't hit the green area.
Your Task: Adjust the delta slider, if possible,
so that the graph does not hit the green region. This will mean that |f(x) - L| < epsilon
for all x such that 0 < |x - a| < delta. Try it with the function above.