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From various applications in electrical and nuclear, I am convinced of the importance of 'time constant' when exponential expressions have to be dealt with. But I am curious to know if the time constant relates to only exponential decay (or rise) or can I extend the understanding to any function which decays and hold an asymptotic value eventually. For let us take a simple example of $\frac{1}{x}$ or $\frac{1}{\sqrt{x}}$.

Assuming there is a physical quantity changing as $\frac{1}{t}$ against time $t$, is it anywhere near logical to say that starting from the beginning at $t=0$ to when $\frac{1}{t}$ takes the value $\frac{1}{e}$, i.e., $t=e$, we get the time constant as $\tau=e$?

Onkar Singh
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    Essential for the exponential function ( $f(x)=a\cdot b^x\ $) is that we have $$f(x+k)=Cf(x)$$ where $C$ is a constant ($b^k$). This behaviour does not occur with other functions, like $\frac{1}{x}$ or $\frac{1}{\sqrt{x}}$ – Peter Sep 08 '20 at 11:14

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