Factorial is not of Exponential Order

Theorem
Let $\Gamma$ denote the gamma function.

Let $f \left({t}\right) = \Gamma \left({t+1}\right) = t!$.

Then:
 * $f$ is not of exponential order.

That is, it grows faster than any exponential.

Proof
From Gamma Function is Continuous on Positive Reals, $f$ is continuous for $t \ge 0$.

Set $t > 0$.

From Stirling's Formula:


 * $\displaystyle t! \sim \sqrt {2 \pi t} \left({\frac t e}\right)^t$

where $\sim$ denotes asymptotic equality.

That is,

As $t^t$ is not of exponential order, it follows that $t!$ is not of exponential order either.