Power Function tends to One as Power tends to Zero/Rational Number

Theorem

Let $a \in \R_{> 0}$.

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

$\map f q = a^q$

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

Then:

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

Proof

Case 1: $a > 1$

If $a > 1$, then:

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

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

Case 2: $a = 1$

If $a = 1$, then:

\(\ds \lim_{x \mathop \to 0} \map f x\)

\(=\)

\(\ds \lim_{x \mathop \to 0} 1^x\)

Definition of $f$

\(\ds \)

\(=\)

\(\ds \lim_{x \mathop \to 0} 1\)

Exponential with Base One is Constant/Rational Number

\(\ds \)

\(=\)

\(\ds 1\)

Real Polynomial Function is Continuous

Case 3: $0 < a < 1$

If $0 < a < 1$, then:

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

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

Hence the result.

$\blacksquare$