Why is it impossible for exponential growth to continue forever?
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It would probably be more accurate to say that exponential growth will eventually exhaust any finite resource no matter how large - and probably sooner than you think.
A typical example would be an exponential growth model of a population of bacteria. Given a large enough supply of nutrients, the number of bacteria in a colony will tend to grow exponentially, but eventually the model will fail, because the bacteria will run out of room or nutrients to grow.
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