Consider the function

Illustrate graphically the epsilon-delta definition of limits to show that f is not continuous at x = 0.

Using Microcalc:

There is a built-in function which will let us construct the function f

  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^2 + 0.002 step(x)
  4. and use the bounds
    x0 = -1 x1 = 1
    y0 = -1 y1 = 1
  5. You will then see the graph of this function. It appears that this function is continuous at x = 0.

  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. Repeat the last two steps several times until you obtain the following graph. The rectangle indicates that there is a problem when choosing epsilon = 0.001