Problem:
Using the TI-85 graphing calculator, graphically determine which of the following functions

y = x7 and y = 4x

has larger values for x large.

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 the F4 key until only y1= appears on the screen.
4. At the prompt y1=, type in

x^7

and press the ENTER key.

5. At the prompt y2=, type in

4^x

and press the ENTER key.

6. The range settings for the graph above is  [0, 10] × [0, 106]:

7. Pick GRAPH from the menu by pressing the F5. Since y = x7 was entered first, it is graphed first:

followed by the graph of y = 4x:

8. It appears that the function y = x7 has larger values.
9. However, this does not continue to happen for x larger. Choose range settings  [10, 15] × [0, 5 × 108]; the graph of y = x7 is

and is followed by the graph of y = 4x:

10. Now it appears that the function y = 4x has larger values. This, in fact, can be verified by using l'Hospital's Rule.

Alternative methods for doing this type of exploration:

1. Consider the graph of x7 - 4x:

with viewing window [0, 50] × [-10, 10].

2. Consider the graph of x7 / 4x:

with viewing window [0, 50] × [0, 2].