1

Given a function of the form $f(x)=\frac{k^x}{x^x}, k>0$ it's easy for me to see convergence to $0$ as $x\to \infty$. But, what about $g(x)=\frac{k^x}{(x^c)^{(x^c)}}$. Unfortunately, I am an Economist by training and don't really know how to handle such limits. I ran some simulations for various $k,c>0$ and found both divergence and convergence. Could someone please tell me how to determine for which values of $k$ and $c$ this converges/diverges as $x\to \infty$? I haven't mentioned the context in which it arose, because it would take far too long and I doubt it would be enlightening - thanks in advance!

EDIT: I failed to mention that of course I'm asking about values of $c\in (0,1)$.

1 Answers1

0

This is not an answer but it is too long for a comment

If in expression $$g(x)=\frac{k^x}{(x^c)^{(x^c)}}$$ you change variable $y=x^c$, you then have to consider the problem of $$f(y)=\frac{k^{y^{\frac{1}{c}}}}{y^y}$$ when $y$ goes to $\infty$ and what seems to be the driving force is the value of $k$. If $k \le 1$, the limit should be $0$ and $\infty$ otherwise.

Could you report cases which do not match this ?

Claude Leibovici
  • 260,315
  • I'm not sure why you say this is not an answer! That change of variable makes it clear to me now. Many thanks! –  Oct 07 '14 at 15:46
  • @GabrielEl-Borhami.I was just meaning that this was too long for a comment. I am glad that you found that the change of variable gave you some idea. Cheers. – Claude Leibovici Oct 07 '14 at 16:11