# Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number

## 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

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

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

$\blacksquare$