What It Means for One Function to Grow Faster
When we compare the rates of growth of two functions, we are asking what happens to their relative sizes as becomes very large. A function grows faster than another function if it eventually becomes much larger than the other one.
For example, compare and . Near the origin, the two functions may have relatively similar values. But as increases, the exponential function eventually becomes dramatically larger than the polynomial.
Calculus gives us a precise way to make this comparison: take the limit of their ratio.
The Basic Test: Compare the Ratio
Suppose we want to compare two positive functions and . Consider the ratio
The limit tells us exactly how the two functions compare as becomes large.
The Growth Hierarchy
For many AP Calculus problems, you can compare functions quickly by remembering the standard hierarchy of growth rates.
Read this from left to right: as becomes very large, each category eventually grows faster than every category to its left.
This gives you a powerful mental shortcut:
- Logarithmic functions grow more slowly than powers of .
- Any positive power of eventually outgrows a logarithm.
- Exponentials eventually outgrow every polynomial.
- Factorials grow even faster than ordinary exponentials.
Worked Examples
Which grows faster as : or ?
Compare the growth of and .
Compare the growth of and .
Which grows faster as : or ?
Compare the growth of and .
Dominant Terms: The Fast Shortcut
When comparing polynomials, you usually do not need to perform a complicated limit. The highest-power term dominates as becomes large.
For example, consider:
The lower-degree terms become insignificant compared with . That means the polynomial behaves more and more like its leading term.
In general:
This is why a fifth-degree polynomial eventually grows faster than every fourth-, third-, second-, or first-degree polynomial.
Same Growth Rate vs. Faster Growth
There is an important distinction between saying that two functions have the same rate of growth and saying that one function grows faster.
If
then the two functions have the same order of growth. They may differ by a constant factor, but neither eventually dominates the other.
For example:
So and have the same growth order.
Why Exponentials Eventually Win
One of the most important ideas to remember is that exponential growth eventually beats polynomial growth.
Even if a polynomial starts out much larger, an exponential eventually catches it and then rapidly pulls away.
The important point is not that the exponential is always larger. It is that there is eventually a point beyond which it remains larger and continues growing much faster.
Using Logarithms to Compare Difficult Functions
Sometimes the functions are complicated enough that direct comparison is difficult. Taking logarithms can turn multiplication, powers, and exponentials into simpler expressions.
For example, consider comparing:
Taking the natural logarithm gives:
Now the comparison is between a term proportional to and a term proportional to . Since grows faster than , the original exponential function eventually dominates.
A Practical AP Calculus Strategy
- Identify the function types. Is each function logarithmic, polynomial, exponential, or something more complicated?
- Look for dominant terms. For polynomials, focus on the highest power.
- Use the growth hierarchy. Remember that logarithms grow more slowly than powers, while exponentials grow faster than powers.
- Form a ratio if necessary. Compare .
- Use limits. A ratio approaching infinity means the numerator grows faster; a ratio approaching zero means the denominator grows faster.
- Use L'Hôpital's Rule when appropriate. If you obtain an indeterminate form such as , differentiation can reveal the relative growth rates.
Common Mistakes
A function can start much larger and still eventually be overtaken. Growth rate concerns what happens as becomes large, not what happens at one particular value of .
If the ratio of two functions approaches a positive finite constant, they have the same order of growth. For example, and grow at the same rate in this sense.
As becomes large, lower-degree terms become insignificant compared with the highest-degree term. For a polynomial such as , the term determines the long-run growth.
The form is indeterminate. Depending on the functions involved, the limit could be 0, a finite constant, infinity, or something else. You must analyze the functions rather than treating infinity like an ordinary number.
Practice Problems
Which grows faster as : or ?
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Evaluate and interpret the result.
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Which grows faster as : or ?
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Evaluate and determine whether the two functions have the same order of growth.
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