Power Function on Base between Zero and 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 $0 < a < 1$.

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

From Ordering of Reciprocals:

$0 < a < 1 \implies 1 < \dfrac 1 a$

So:

\(\ds \lim_{r \mathop \to 0} \paren {\frac 1 a}^r\) \(=\) \(\ds 1\) Power Function on Base greater than One tends to One as Power tends to Zero: Rational Number
\(\ds \leadsto \ \ \) \(\ds \lim_{r \mathop \to 0} \frac 1 {a^r}\) \(=\) \(\ds 1\) Power of Quotient: Rational Numbers
\(\ds \leadsto \ \ \) \(\ds \frac 1 {\ds \lim_{r \mathop \to 0} a^r}\) \(=\) \(\ds 1\) Quotient Rule for Limits of Real Functions
\(\ds \leadsto \ \ \) \(\ds \lim_{r \mathop \to 0} a^r\) \(=\) \(\ds 1\) taking reciprocal of each side

$\blacksquare$

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