In this Calculus question we need to find the equation of a function $f(x)$ that satisfies these 2 properties based on limits:
1) for all $b > 1$, $\lim\limits_{x \to \infty} \frac{f(x)}{x^b} = 0$
2) for all $b \leq 1$, $\lim\limits_{x \to \infty} \frac{f(x)}{x^b} = \infty$
Initially, i did not think it was too bad, but then when I tried out functions, elementary functions, I had difficulties.
I am thinking that its probably an exponential type of function. I was thinking possibly the gaussian function, but i think it fails the second condition. Another function is possibly root i think, but not fully certain. That is all i could think up of, hit a brick wall with this one.
Basically you want $f(x)$ grows "little bit" faster than $x$. So you can make $x$ a coefficient that grows slower than all $x^a$. That's "little bit". For example, how about $\log x$? Put $f(x) = x\log x$ in and test it.