# 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 |

$\blacksquare$