Graphically determine which of the following functions

y = x7 and y = 4x

has larger values for x large.

Using Maple:

> plot(x^7,x=0..10,thickness=2);

> plot({x^7,4^x},x=0..10,y=0..10^7,thickness=2);

From the two graphs above, it appears that y = x7 is the answer. However, if we make x larger then we see the following graph which leads us to a different conclusion. This, in fact, can be verified by using l'Hospital's Rule.

> plot({x^7,4^x},x=0..15,y=0..10^8,thickness=2);

Two alternative methods for doing this type of investigations are to consider the graph of y = x7 - 4x:

> plot(x^7-4^x,x=0..50,y=-10..10,thickness=2);

or the graph of y = x7 / 4x:

> plot(x^7/4^x,x=0..50,y=0..2,thickness=2);