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The first part of this has nothing to do with my question. It was just something I was scribbling:

Let's say I have a population that's growing. If p is my population:

$\frac{p'(t+1)}{p'(t)} >1$

If I want to check the rate as the difference approaches 0, then

$\lim_{i\to0} \frac{p'(t+i)}{p'(t)} = 1$

Well this is explicitly saying that the first function is absolutely greater than 1. But, if you add even the most tiny of values to t, then the limit is definitely equal to 1.

There's no question here, but I just thought it was interesting to see the necessity of a change of a symbol, by adding an infinitesimal. I don't know if that makes sense, I just thought it was interesting.

I better add a question just to make this post legit. Is math love? or how about: Is the level of emigration in a country proportional to the size of the population?

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I expect you want $\dfrac{p(t+1)}{p(t)}\gt 1$. Your inequality seems to say that the rate of growth of the population is growing. –  André Nicolas Sep 11 '12 at 0:10
    
Yes that's what I meant. the rate of growth is accelerating. –  Korgan Rivera Sep 11 '12 at 0:11

2 Answers 2

Your first equation isn't saying what you say in words. If the population is growing, you would expect $\frac {p(t+1)}{p(t)} \gt 1$, but the rate of growth could be slowing. It might be easier to say $p(t+1)-p(t) \gt 0$. Then if the population is leveling off, you could say $\lim_{t \to t_m} p(t+1)-p(t) = 0$, were $t_m$ is the time it hits steady state (could be $\infty$, but might not be). I don't see where there is any need for an infinitesimal, nor how you are using it.

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When we are trying to use mathematics to deal with a real phenomenon, we make a mathematical model. It is unreasonable to expect a model to fit perfectly, or even very well. As a rough first approximation, when times are stable, one would expect that the number of people who emigrate is proportional to the population, with the proportionality constant varying substantially from country to country.

But one can expect the proportionality "constant" to change as economic and political conditions change, either in the country people emigrate from, or in the favourite host country. So assuming proportionality is undoubtedly only a crude short-term model.

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