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

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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.