In this series of posts, I consider how two different definitions of the number are related to each other. The number is usually introduced at two different places in the mathematics curriculum:
- Algebra II/Precalculus: If dollars are invested at interest rate for years with continuous compound interest, then the amount of money after years is .
- Calculus: The number is defined to be the number so that the area under the curve from to is equal to , so that
These two definitions appear to be very, very different. One deals with making money. The other deals with the area under a hyperbola. Amazingly, these two definitions are related to each other. In this series of posts, I’ll discuss the connection between the two.
In yesterday’s post, I proved the following theorem, thus completing a long train of argument that began with the second definition of and ending with a critical step in the derivation of the continuous compound interest formula. Today, I present an alternate proof of the theorem using L’Hopital’s rule.
Proof #2. Let’s write the left-hand side as
Let’s take the natural logarithm of both sides:
Since is continuous, we can interchange the function and the limit on the right-hand side:
The right-hand side follows the indeterminate form , as so we may apply L’Hopital’s Rule. Taking the derivative of both the numerator and denominator, we find
Therefore, the original limit is .