Why Matters
The number is one of the most important constants in mathematics. Its approximate value is , but memorizing that decimal does not explain why appears everywhere in calculus.
The deeper reason is that is the natural base for continuous exponential growth. It is built into the mathematics of continuously compounded interest, population growth, radioactive decay, differential equations, and many other models.
In AP Calculus, one of the most important ways to understand is through its limit definition.
The Fundamental Limit Definition
The number can be defined by the limit:
This says that as becomes very large, the expression approaches the number .
You should recognize as a limit that approaches , not as an expression that can simply be evaluated by plugging in infinity. The limit is describing what happens as grows without bound.
Where Does This Limit Come From?
One way to see why the limit is connected to is through compound interest. Suppose you invest at an annual interest rate of 100%, but instead of adding the interest only once at the end of the year, you compound it times during the year.
Each compounding period multiplies your money by . After compounding periods, your balance is:
Now imagine increasing the number of compounding periods. As gets larger and larger, the balance approaches a fixed number: .
This is the mathematical origin of the constant in this particular model of continuous growth.
Equivalent Forms of the Limit
You may encounter several equivalent forms of the limit definition. The most common is:
By substituting , so that , we obtain:
This gives another extremely useful form:
The two forms describe the same fundamental behavior. You should be comfortable recognizing either one.
Worked Examples
Evaluate .
Evaluate .
Evaluate .
Evaluate .
Connecting the Limit Definition to the Derivative of
The importance of becomes even clearer when we look at the derivative of the exponential function.
Using the definition of a derivative for :
Factor out :
Now the limit-definition form of appears naturally:
Since that limit equals 1, we obtain the remarkable result:
This is one of the reasons is so special. The exponential function is its own derivative.
Common Mistakes
You cannot simply substitute into . Infinity describes an unbounded process, not a numerical value that can be plugged into an expression.
The expressions and are related but not identical. The first is the standard limit definition. The second is an equivalent form obtained through a substitution.
When you see a limit such as , setting often converts it directly into the standard definition. Always check whether a substitution can expose the familiar pattern.
Knowing that is approximately 2.718 is useful, but AP Calculus problems often test whether you understand why appears and how its defining limit can be transformed.
Practice Problems
Evaluate .
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Evaluate .
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Evaluate .
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Explain why represents the number .
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