Function with Limit at Infinity of Exponential Order Zero

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Theorem

Let $f: \hointr 0 \to \to \R$ be a real function.

Let $f$ be continuous everywhere on their domains, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$.



Let $f$ have a (finite) limit at infinity.


Then $f$ is of exponential order $0$.


Proof

Denote $\displaystyle L = \lim_{t \mathop \to +\infty} \map f t$.

Define the constant mapping:

$\map C t = - L$

Further define:

$\map g t = \map f t + \map C t$

From:

Constant Function is of Exponential Order Zero,
Sum of Functions of Exponential Order,

it is sufficient to prove that $g$ is of exponential order $0$.


Fix $\epsilon > 0$ arbitrarily small.

By definition of limit at infinity, there exists $c \in \R$ such that:

$\forall t > c: \size {\map f t - L} < \epsilon$


Therefore:

\(\, \displaystyle \forall t \ge c + 1 : \, \) \(\displaystyle \size {\map g t}\) \(=\) \(\displaystyle \size {\map f t + \map C t}\)
\(\displaystyle \) \(=\) \(\displaystyle \size {\map f t - L}\)
\(\displaystyle \) \(<\) \(\displaystyle \epsilon\)
\(\displaystyle \) \(=\) \(\displaystyle \epsilon \cdot e^0\) Exponential of Zero

The result follows from the definition of exponential order, with $M = c + 1$, $K = \epsilon$, and $a = 0$.

$\blacksquare$