Imagine we have two different differentiable 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 ?


It means that after a long time the rate of change $\dot{f}(t)$ of a quantity $f(t)$ becomes proportional to the rate of change $\dot{g}(t)$ of another quantity $g(t)$. An example of this kind of behaviour could be the following.

Imagine two runners on a track. Denote their instantaneous speeds as $\dot{f}(t)$ and $\dot{g}(t)$, respectively. They start next to each other, at the same time. Runner 1 has a (constant) natural running speed of $v$, while that of runner 2 is $u = c \cdot v$.

But runner 1 is also an overenthusiastic starter. He always begins at a speed $\dot{f}(t)|_{t=0} > v$, but gradually has to slow down because he cannot keep this up. On the other hand, runner 2 is a slow starter. His speed is $\dot{g}(t)|_{t=0} < u$ in the beginning, but he gradually gets faster and faster, tending towards his natural pace.

This is a system that behaves according to your limit. After a sufficient amount of time has passed, the speed of runner 1 will tend to $v$ and that of runner 2 to $u$, meaning that $||\dot{f}(t)||/||\dot{g}(t)||$ will tend to $v/u = c$ for such long times.

  • $\begingroup$ I see, it seems like the relative steady speed of each runner. Thanks! $\endgroup$ – user96212 Oct 3 '13 at 18:31

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