has larger values for x large.

[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 = 4^x:

   

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 = 4^x:

   

10. Now it appears that the function y = 4^x 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 - 4^x:
   

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

2. Consider the graph of x⁷ / 4^x:
   

   with viewing window [0, 50] × [0, 2].