Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number

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Theorem

Let $a \in \R_{> 0}$ be a strictly positive real number such that $a > 1$.

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

$\map f r = a^r$

where $a^r$ denotes $a$ to the power of $r$.


Then:

$\ds \lim_{r \mathop \to 0} \map f r = 1$


Proof

We start by treating the right-sided limit.

Let $0 < r < 1$.


Lemma

Let $a \in \R$ be a real number such that $a > 1$.

Let $r \in \Q_{> 0}$ be a strictly positive rational number such that $r < 1$.


Then:

$1 < a^r < 1 + a r$

$\Box$


From the lemma:

$1 < a^r < 1 + a r$

Also:

\(\ds \lim_{r \mathop \to 0} \paren {1 + a r}\) \(=\) \(\ds 1 + a \times 0\) Real Polynomial Function is Continuous
\(\ds \) \(=\) \(\ds 1\) Real Zero is Zero Element
\(\ds \) \(=\) \(\ds \lim_{r \mathop \to 0} 1\) Real Polynomial Function is Continuous

So from the Squeeze Theorem for Functions:

$\ds \lim_{r \mathop \to 0^+} a^r = 1$


We now treat the left-sided limit.

Let $-1 < r < 0$.

\(\ds -1\) \(<\) \(\, \ds r \, \) \(\, \ds < \, \) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds 0\) \(<\) \(\, \ds -r \, \) \(\, \ds < \, \) \(\ds 1\) Order of Real Numbers is Dual of Order of their Negatives
\(\ds \leadsto \ \ \) \(\ds 1\) \(<\) \(\, \ds a^{-r} \, \) \(\, \ds < \, \) \(\ds 1 - a r\) Lemma
\(\ds \leadsto \ \ \) \(\ds 1\) \(<\) \(\, \ds \frac 1 {a^r} \, \) \(\, \ds < \, \) \(\ds 1 - a r\) Real Number to Negative Power: Rational Number
\(\ds \leadsto \ \ \) \(\ds \frac 1 {1 - a r}\) \(<\) \(\, \ds a^r \, \) \(\, \ds < \, \) \(\ds 1\) Ordering of Reciprocals


Also:

\(\ds \lim_{r \mathop \to 0} \dfrac 1 {1 - a r}\) \(=\) \(\ds \frac 1 {\ds \lim_{r \mathop \to 0} \paren {1 - a r} }\) Quotient Rule for Limits of Real Functions
\(\ds \) \(=\) \(\ds \frac 1 {1 - a \times 0}\) Real Polynomial Function is Continuous
\(\ds \) \(=\) \(\ds 1\) Real Zero is Zero Element
\(\ds \) \(=\) \(\ds \lim_{r \mathop \to 0} 1\) Real Polynomial Function is Continuous

So from the Squeeze Theorem for Functions:

$\ds \lim_{r \mathop \to 0^-} a^r = 1$

From Limit iff Limits from Left and Right:

$\ds \lim_{r \mathop \to 0} a^r = 1$

Hence the result.

$\blacksquare$

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