When is $\lim_{t \to \infty} \frac{f(t)}{t}$ equal to $\lim_{t \to \infty} \frac{df}{dt}$? Under what circumstances are $\lim_{t \to \infty} \frac{f(t)}{t}$ and $\lim_{t \to \infty} \frac{df}{dt}$ equal?
I'm aware that in some cases one or the other limit may not exist (e.g. if $f(t) = sin(t)$ the limit of the derivative does not exist.)
More precisely, I'm interested in whether  the limits are equal if $f$ is monotonically increasing and smooth.

$f(x) = x + \sin(x)$ is monotone, as $f'(x) \geq 0\ $ for all $x.$
This function satisfies $\ \lim_{t \to \infty} \frac{f(t)}{t} = 1\ $ but $\ \lim_{t \to \infty}\frac{df}{dt}\ $ does not exist.
The question of: "when are they equal" is more interesting and difficult.
The above example of $f(x) = x + \sin(x)$ shows that, in general, we do not have:
$  \lim_{t \to \infty} \frac{f(t)}{t} = c,\implies\ \lim_{t \to \infty}\frac{df}{dt}=c\ $.
The accepted answer is clever and tells us about the converse statement.
