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Euler's number

The Euler number e = 2.7182… is the limit

 

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Explanation

We start with the exponential function

f (x) = a^x

Substitution in the definition of the derivative gives

and so

For x = 0 this becomes

That is a constant, as there is no x in it. The value depends only on a, the base. You can therefore write

We want to know, for what value of a the function f ′(0) = 1, because then is f ′(x) = a^x, and the function is it's own derivative. So we start the calculation of

Multiplcation with h

Shuffling

Exponentiation of both sides

This is

Let , and because h → 0 gives this n → ∞, and so

 

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History The mathematician Leonhardt Euler (1707 - 1783) is famous for his contributions on this constant.

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