Using the TI-85 graphing calculator, find graphically the delta that corresponds to

[Press here to see animation again!]

- 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.
- Press the GRAPH key and then pick
**y(x)=**by pressing the F1 key. - If necessary, keep pressing F4 key until only
**y1=**appears on the screen. - Type in the piecewise defined function
**x^(1/3)**and press the ENTER key.

- Press EXIT and the F2 keys
to pick
**RANGE**from the menu. - Let us begin with
**e = 0.1**and try to find a corresponding**d**; our first guess will be**d = 0.1**. - We enter the viewing window
**[-d, d] × [-e, e]**: - Press the F5 key to graph the function:
and it is clear that

**d**is too large. - We now enter
**d = 0.01**: - Press the F5 key to graph the function:
and it is clear again that

**d**is too large. - We now try
**d = 0.001**: - 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**|x**.^{1/3}- 0| < e = 0.1 - 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**: - 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**|x**.^{1/3}- 0| < e = 0.01