find $f(n)$ that isn't little $o(n^2)$ and isn't $w(n)$

Question

im having a problem with a question that tells me to find $f(n)$ so that

$f(n) \neq o(n^2)$ meaning also that $\lim_{\rightarrow\infty} \frac{f(n)}{n^2} \neq 0$

and $f(n) \neq w(n)$ meaning also that $\lim_{n\rightarrow\infty} \frac{f(n)}{n} \neq \infty$

I have tried solving each part and the first equation tells me that if $f(n) = x^q$ then $2\leq q$ and the second tells me that $q\leq 1$.

is there even a solution to this problem?

Answer 1

A function which oscillates back and forth between $x$ and $x^2$ will work.

Explicitly, consider the function $f$ given by $f(x)=x\,\sin^2(x)+x^2\,\cos^2(x)$.