Problem:
Illustrate graphically the epsilon-delta definition of limits to obtain evidence that


Visualization:
Using Microcalc:

  1. Choose Beginning Calculus from the initial menu and press .
  2. Use the arrow keys to move to the menu item Graph y = F(x), press and .
  3. At the prompt F(x) =, enter the function
    x^(1/3)
  4. and use the bounds
    x0 = -1 x1 = 1
    y0 = -1 y1 = 1
  5. You will then see the graph of this function.
  6. Press and use the arrow keys to move to Zoom In. Press and .
  7. You will see the following graph with a rectangle (really a square!) which encloses the area to which the zoom command will take you. This rectangle corresponds to choosing epsilon = 0.1 and delta = 0.1.

  8. Press and the zoomed in graph is drawn. Clearly, delta = 0.1 does not work.
  9. We'll make the delta smaller by changing the x coordinates. Press and use the arrow keys to Change [x0,x1] and press .
  10. Now use the bounds
    x0 = - 0.0005 x1 = 0.0005
  11. From the following picture, we see that this choice of delta works.