Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Extension of Rational Exponential

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Theorem

The following definition of the concept of the real exponential function:

As the Limit of a Sequence

The exponential function can be defined as the following limit of a sequence:

$\exp x := \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$


implies the following definition:

As an Extension of the Rational Exponential

Let $e$ denote Euler's number.

Let $f: \Q \to \R$ denote the real-valued function defined as:

$\map f x = e^x$

That is, let $\map f x$ denote $e$ to the power of $x$, for rational $x$.


Then $\exp : \R \to \R$ is defined to be the unique continuous extension of $f$ to $\R$.

$\map \exp x$ is called the exponential of $x$.


Proof

Let the restriction of the exponential function to the rationals be defined as:

$\ds \exp \restriction_\Q: x \mapsto \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$


Thus, let $e$ be Euler's Number defined as:

$e = \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac 1 n}^n$


For $x = 0$:

\(\ds \exp \restriction_\Q \paren 0\) \(=\) \(\ds \lim_{n \mathop \to +\infty} \paren {1 + \frac 0 n}^n\)
\(\ds \) \(=\) \(\ds 1\)
\(\ds \) \(=\) \(\ds e^0\)

For $x \ne 0$:

\(\ds \map {\exp \restriction_\Q} x\) \(=\) \(\ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n\)
\(\ds \) \(=\) \(\ds \lim_{\paren {n / x} \mathop \to +\infty} \paren {\paren {1 + \frac 1 {\paren {n / x} } }^{\paren {n / x} } }^x\) Exponent Combination Laws
\(\ds \) \(=\) \(\ds e^x\)

where the continuity in the last step follows a fortiori from Power Function to Rational Power permits Unique Continuous Extension.


For $x \in \R \setminus \Q$, we invoke Power Function to Rational Power permits Unique Continuous Extension.

$\blacksquare$


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Sources

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