Problem:
Using the TI-86 graphing calculator, investigate numerically the behavior of the function

for values of x near 0.


Visualization:

  1. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
  2. If necessary, keep pressing the F4 key until only y1= appears on the screen.
  3. Be sure that PLOTS 1, 2 and 3 are turned-off.
  4. At the prompt y1=, type in

    ((x^3+8)^(1/3)-2)/x^3

    and press the ENTER key.

  5. Go to the WINDOW menu by pressing 2nd M2.
  6. The WINDOW settings for the initial graph is  [-1, 1] × [-1, 1]:

  7. Pick GRAPH from the menu by pressing the F5 key.
  8. Pick ZOOM from the menu by pressing the F3 key.
  9. Press the MORE key twice so that ZFACT appears on the menu.
  10. Press the F2 key to pick ZFACT.
  11. Enter 10 at each of the prompts xFact= and yFact= followed by the ENTER key. With these settings, the zooming in will be at a factor of 1/10. Since we started with an interval of length 1, the length of the interval after one zooming in is 0.1; after two zooming in's, the length is 0.01, etc.
  12. Pick ZOOM by pressing the F3 key again.
  13. Pick ZIN from the menu by pressing the F2 key.
  14. You will now see a flashing tick mark in the middle of the screen and the coordinates of the tick mark at the bottom of the screen. Using the arrow keys, move the tick mark as close as possible to the intersection of the graph with the vertical line which is the y-axis and press the ENTER key.
  15. Repeat the latter step a couple of times to see the following:

    This graph indicates that there may be a problem.

  16. One more zoom in and we get the graph:

    It appears that the limit does not exist. However, this problem is due to round off error and the limit, in fact, does exist!

    This page was written with the help of Chris O'Brien.