Problem:
Using the TI-85 graphing calculator, find graphically the delta that corresponds to e = 0.1 and e = 0.01 in the definition of limits to obtain evidence that

Visualization:

[Press here to see animation again!]

1. If you are not familiar with the graphing of functions on the TI-85, then first read the Initial Setup page from Little's Basic Guide to the TI-85.
2. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
3. If necessary, keep pressing F4 key until only y1= appears on the screen.
4. Type in the piecewise defined function

x^(1/3)

and press the ENTER key.

5. Press EXIT and the F2 keys to pick RANGE from the menu.
6. Let us begin with e = 0.1 and try to find a corresponding d; our first guess will be d = 0.1.
7. We enter the viewing window [-dd] × [-ee]:

8. Press the F5 key to graph the function:

and it is clear that d is too large.

9. We now enter d = 0.01:

10. Press the F5 key to graph the function:

and it is clear again that d is too large.

11. We now try d = 0.001:

12. Press the F5 key to graph the function:

and it appears that d = 0.001 works; i.e., 0 < |x - 0| < d = 0.001 implies |x1/3 - 0| < e = 0.1.

13. Let us try a different e, say e = 0.01; there are several possibilities for the choice of d. Note that the previous d is the cube of e; so let us try d = 0.000001 which is the cube of 0.01:

14. Press the F5 key to graph the function:

and again it appears that d = 0.000001 works; i.e., 0 < |x - 0| < d = 0.000001 implies |x1/3 - 0| < e = 0.01.