Task
The purpose of this exploration is to help you develop your intuition about how common functions compare as they approach infinity. While there are many functions that we could compare, the ones that you need to compare as part of this exploration are the following:
Exponential (e. g. ex)
Factorial (e. g. x!)
Logarithmic (e. g. ln x)
Polynomial (e. g. x5 – 3x2 + 100)
In addition to developing your intuition about some common functions in this unit, the ideas in this exploration are very important in computer science. Mathematicians and computer scientists analyze algorithms to see if a computer has the capacity to solve them or not. We will not say more about this now, but if you are interested in the practical application of analyzing the end behavior of a given function, then look up the traveling salesman problem. It is a good introduction to this branch of mathematics.

Defining the Race:
The first question that we need to answer is how should we define a function as the winner of a race to infinity? Before going on, take a few minutes and think about how you would define the winner.
In this lesson, we will say that a function “wins” the race if at some point it passes another function, and the other function never passes it after that. Keep in mind that under different circumstances, we may be more interested in finding functions that grow slower. It isn’t always the case that the fastest is best.

Picking a Function
Now that we have a common way to define the winner of the race, it is time for you to create the function that you think will win. There is only one rule: your function has to be either exponential, factorial, logarithmic, or polynomial. You can’t mix functions (e. g. lnx!2).