Growth of a function

disregardthat

Science Advisor
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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?
 
906
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.
 

disregardthat

Science Advisor
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Thanks,

[tex]x^{\sqrt{x}}[/tex] is such a function.
 
906
2
Or [tex]e^{\sqrt{x}}[/tex], or [tex]e^{x/\ln{x}}[/tex].
 

disregardthat

Science Advisor
1,842
33
I'm sure we can find many as you pointed out.
 

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