# The physical interpretation of limit of ratio of two functions

Imagine we have two differential functions $f(t)$ and $g(t)$ where $t$ generally represents the time, if there exists the following limit as $$\lim\limits_{t\rightarrow \infty } \frac{\| \dot{f}(t) \|}{\|\dot{g}(t)\|}=c$$ Then, is there any appropriate physical explanation for this limit ? If ignore the limit, the $\frac{\| \dot{f}(t) \|}{\|\dot{g}(t)\|}$ is sort of instantaneous reletive absolute rate of change of $f(t)$ over $g(t)$, but what is that when limit involves in ?

If $f(t)$ denotes the position of a particle at time $t$, then $\|\dot f(t)\|$ is the speed of the particle at that time. Your limit existing roughly means that the particle represented by $f$ travels $c$ times as fast (ignoring the direction of the velocity) as the one represented by $g$ when $t$ is large.