A Delta-Epsilon Applet

This Applet lets the student work with the Delta-Epsilon definition of limit.

Recall that when we say the limit as x approaches a of f(x) is L we mean that for every epsilon > 0 there is a delta > zero with
|f(x) - L| < epsilon whenever 0 < | x-a | < delta.

This definition is often hard for students to understand, so it is useful to do a visual translation -
The limit as x approaches a of f(x) is L means for every height epsilon we can find a width delta so that for a box centered at (a,L) going up and down epsilon and right and left delta, the function only escapes through the sides of the box rather than the top and bottom.

Proving a limit means finding a rule that finds a delta for every positive epsilon and proving that it works.  For functions of the level that we typically see in a calculus class, it is pretty convincing to find a delta when epsilon is .1, .01, and .001, and to use a picture to show that the deltas work.

Notes on use of the applet:

Suggested exercises:
  1. The first example is an easy limit.  Find a delta corresponding to epsilon values of .1, .01, and .001. 
  2. By the time you have found the delta for epsilon = .001, you have zoomed in far enough that the graph should look like a straight line.  Use this fact to come up with a rule for small enough epsilons.
  3. Look at examples 2 and 3.  Note that the function is not defined at x=a, but the whole is impossible to see on the graph.
  4. Example 4 is typical of when we have to check continuity since the function is piecewise defines.  What is the other value of x that should be checked for continuity.
  5. Examples 5 and 6 are piecewise defined functions that are not continuous.  Find a value of epsilon for which there is no positive value of delta that works.  (Epsilon needs to be small enough that the right hand limit and left hand limit are more than 2 epsilon apart.)
  6. Use the applet to explore the function y = sin(1/x).  Explain why the function has no limit as x approaches 0.
  7. Use the applet to explore problems from your textbook.

This applet was designed as part of David Eck's "Java Components For Mathematics" project. Minor modifications were made locally.

Return to local JCM Applet page.

Maintained by Mike May, S.J.  Modified 9/4/04