This Applet lets the student work with the Delta-Epsilon definition of
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:
Dragging on the graph produces a box for zooming in.
Functions can be loaded either from the drop down menu at the top
of the applet, or typed in at the bottom of the applet.
the values of a and L can be controlled either by slider or by
typing in the text fields.
The first example is an easy limit. Find a delta
corresponding to epsilon values of .1, .01, and .001.
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
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.
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.
Examples 5 and 6 are piecewise defined functions that are not
continuous. Find a value of epsilon for which there is no
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.)
Use the applet to explore the function y = sin(1/x).
Explain why the function has no limit as x approaches 0.
Use the applet to explore problems from your textbook.