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Growth of a function

  1. Nov 28, 2009 #1

    disregardthat

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    Does there exist a continuous function which outgrows polynomial growth, but not exponential growth?

    I.e. does a there exist a continuous function f such that [tex]\frac{x^n}{f(x)} \to 0[/tex] and [tex]\frac{f(x)}{a^x} \to 0[/tex] for all positive real n and a?
     
  2. jcsd
  3. Nov 28, 2009 #2
    Yes. Look at it this way: you take logs of the polynomial and the exponential, you get [tex]g_1(x) = C_1 \ln x[/tex] and [tex]g_2(x) = C_2 x[/tex]. Can you find a function that grows faster than g_1 and slower than g_2 for all C? Clearly you can, because ln grows extremely slowly.
     
  4. Nov 28, 2009 #3

    disregardthat

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    Thanks,

    [tex]x^{\sqrt{x}}[/tex] is such a function.
     
  5. Nov 28, 2009 #4
    Or [tex]e^{\sqrt{x}}[/tex], or [tex]e^{x/\ln{x}}[/tex].
     
  6. Nov 29, 2009 #5

    disregardthat

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    I'm sure we can find many as you pointed out.
     
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