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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 ?

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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.

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  • $\begingroup$ Got it, have to recall the definition of limit of function, thanks! $\endgroup$ – user96212 Oct 2 '13 at 23:16

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