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.

If $\lim_{x \to \infty}f(x) = \pm \infty$ and $\lim_{x \to \infty}f′(x)$ exists, then you have equality by L'Hopital’s rule.
