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the Number e as a Limit

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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

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

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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,

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

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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

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

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

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LEONHARD EULER (1707 - 1783)

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