Is exponential growth and decay faster than polynomial growth and decay?I know the answer is yes for growth conditions, but I don't see how it's obvious that exponential decay is faster than polynomial decay, say for a polynomial x 2 .

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Is exponential growth and decay faster than polynomial growth and decay?I know the answer is yes for growth conditions, but I don&#039;t see how it&#039;s obvious that exponential decay is faster than polynomial decay, say for a polynomial       x    2  .

micelarnyiz · Accepted Answer

he most intuitive way of thinking about it is by considering      e    x    =      1          0      !        +      x          1      !        +            x      2              2      !        +            x      3              3      !        +  ⋯Given any   p  ∈      R  , it&#039;s easy to see that       x    k    =      x          ⌈      p      ⌉       grows faster than       x    p   as x tends to infinity. But now, whatever k is, there is a term             x              k        +        1                    (      k      +      1      )      !      ! in the expansion of       e    x  , so       e    x   must grow faster than       x    p   for any p.Similarly, if we can consider the case for x tending to infinity in       e          −      x       and see whether that goes to zero faster than a polynomial grows. We simply note that       e          −      x        =      1          e      x      . Hence as       e    x   grows faster than any polynomial,       1          e              x             must decay faster than any polynomial.

Is exponential growth and decay faster than polynomial growth and decay? I know the answer is yes for growth conditions, but I don't see how it's obvious that exponential decay is faster than polynomial decay, say for a polynomial x^2.

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