We now consider a very special number in mathematics. In 1741, Leonhard Euler named this number e. Though you may not have encountered it before, you will see here and in future mathematics courses that it has many important applications. To explain this number, we use the compound interest formula $A=P{(1+r/n)}^{nt}$ discussed in Example 4. Suppose that $1 is invested at 100% interest for 1 year. Since $P=1$, $r=100\%=1$, and $t=1$, the formula above becomes a function A defined in terms of the number of compounding periods n:

Let’s visualize this function using its graph, shown at left, and explore the values of $A(n)$ as $n\to \infty$. Consider the graph for larger and larger values of n. Does this function have a horizontal asymptote?

Let’s find some function values using a calculator.

It appears from these values that the graph does have a horizontal asymptote, $y\approx 2.7$. As the values of n get larger and larger, the function values get closer and closer to the number Euler named e. Its decimal representation does not terminate or repeat; it is irrational.

Example 5

Find each value of $e^x$, to four decimal places, using the key on a calculator.

1. $e^3$

2. $e^{-0.23}$

3. $e^0$

Solution

	FUNCTION VALUE	READOUT	ROUNDED
a)	$e^3$		20.0855
b)	$e^{-0.23}$		0.7945
c)	$e^0$		1

Now Try Exercises 1 and 3.