Power the race horse, Polly the parrot, and Log the log walk...or roll...into a bar. After a couple root beers, Log challenges the others to a race on a horse track. Who will we bet on? Power, of course! Why would a log challenge a race horse to a race, anyway?
Some functions grow more quickly than others, and this section will help us determine which functions will dominate and which functions will lag behind. It'll come in handy for evaluating limits of some pretty gnarly functions.
In the world of horse races, power functions like 2x will always grow faster than plain old polynomials, no matter how high the degree of the polynomial. By "grow faster'' we mean that if we go far enough to the right on the graph, the power function will be on top of the polynomial. We also mean that
If a function has a power function term in it, we consider it a power function for now.
Now we'd better correct a little something: we could have a power function getting more and more negative instead of more positive. We could also have a polynomial getting more negative instead of positive. It's important to take signs into consideration when determining this limit, because we could have
Who wins when we compare polynomials and logarithmic functions? Look at a picture.
Eventually, after not too long, the polynomial will pull ahead of the logarithmic function. This makes sense, because the polynomial is curved upwards, while the logarithmic curve looks like it's flattening out.
The logarithmic curve never flattens out; it has no horizontal asymptotes, and grows without bound, but it does so slowly.
If we take some limits, we find
Think of these three types of functions as if they are racing. Power functions are like powerful race horses; polynomials (Polly want a cracker?) are like parrots fluttering along; and logarithmic functions are like logs, plodding and slow. The category a function belongs to is determined by its leading term, that is, the individual term that "grows the fastest."