1. the Number e as a Limit

2. The Number e as a Limit by M. Seppälä The mathematical constant e is that number for which the tangent to the graph of e x at x = 0 has the slope 1. e 2.718281828 Exponentials

3. The Number e as a Limit by M. Seppälä Eulers Argument For any given positive number x, is infinitely large. For infinitely small

4. The Number e as a Limit by M. Seppälä The Number e as a Limit A consequence of the definition of the mathematical constant e was that D(e x ) = e x. By the Inverse Function Rule, this implies that D(ln x) = 1/x. In particular,

5. The Number e as a Limit by M. Seppälä The Number e as a Limit implies The formula

6. The Number e as a Limit by M. Seppälä Since ln is a continuous function, This implies We have shown that. THE NUMBER E AS A LIMIT

7. The Number e as a Limit by M. Seppälä Equivalently: We have shown that THE EXPONENTIAL FUNCTION

8. The Number e as a Limit by M. Seppälä Hence THE EXPONENTIAL FUNCTION

9. LEONHARD EULER (1707 - 1783)