Identity is of Exponential Order Epsilon
Jump to navigation
Jump to search
Theorem
Let $I_\R: t \mapsto t$ be the identity mapping on $\R_{\ge 0}$.
Then $I_\R$ is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.
Proof
\(\ds e^{\epsilon t}\) | \(\ge\) | \(\ds 1 + \epsilon t\) | Exponential of $t$ not less than $1 + t$ | |||||||||||
\(\ds \) | \(>\) | \(\ds \epsilon t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds K e^{\epsilon t}\) | \(>\) | \(\ds t\) | $K = \dfrac 1 \epsilon$ |
$\blacksquare$