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:

$f \left({r}\right) = a^r$

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


Then:

$\displaystyle \lim_{r \mathop \to 0} f \left({r}\right) = 1$


Proof

From Ordering of Reciprocals:

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

So:

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

$\blacksquare$