How to ensure when the derivative approaches zero, the integral approaches a constant?

Question

I asked a very similar question here. But now this is different. Suppose $f(t)$ is differentiable and $c$ is a finite constant, then the following statement looks correct, but in fact it is not: \begin{equation} \lim\limits_{t \to \infty} f'(t) = 0 \implies \lim\limits_{t \to \infty} f(t)=c \end{equation} A counter-example is $f(t)=\ln(t)$. Now the question is, what condition should be used for the above statement to be true?

Answer 1

An equivalent condition is that $$ \int_a^\infty f'(t)\,dt $$ converges as an improper Riemann integral for some $a\in\Bbb R$. This happens for instance if $|f'(x)|\le C\,x^{-p}$ for some $C\ge0$ and $p>1$.

Answer 2

One way you could potentially change your statement so that it is true is to assert that the function is bounded asymptotically (as $t \rightarrow \infty$).

If $\lim\limits_{t \rightarrow \infty} f'(t) = 0$ and $\exists n,M$ such that $\forall t > n, |f(t)| \le M$, then $\exists c$ such that $\lim\limits_{t \rightarrow \infty} f(t) = c$

I don't have a proof of this currently, but I believe it should be true, and in fact I believe that this also works in reverse. I will provide a proof if I can come up with one.

Answer 3

Two ways for a differentiable (and hence, continuous) function $f(t)$ to satisfy $\lim_{t \to \infty} f(t) = c$:

(1) $f(t) = c$; i.e., is a constant function

(2) $f(t)$ has a horizontal asymptote $y=c$ in the positive x-direction.

Hence, restrict $f(t)$ to be either of these two function types and your original assertion $\lim_{t \to \infty} f'(t) \Rightarrow \lim_{t \to \infty} f(t)$ will be true.