Power Function tends to One as Power tends to Zero/Rational Number
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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.
$\blacksquare$