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It would make sense that there's always a solution, since I'm pretty sure that equations of the form $a^{x}$ diverge faster than $x^{a}$, but does anyone have a proof?

EDIT: Whoops, sorry, I meant $P(x)-{b}^{x}+1$.

EDIT2: I really fell short on this question. I also meant $b>1$

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    Thank you! That makes sense.2012-12-09

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Counterexample: $P(x)=x+1$, $b=2$.

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    Whoops, sorry, I made a dumb mistake. I meant $+1$ instead of $-1$. Basically, I'm trying to show that exponential functions always diverge faster than polynomials.2012-12-08