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The Number e as a Limit. 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.

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Presentation on theme: "The Number e as a Limit. 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."— Presentation transcript:

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 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 ( )


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