Avoiding the projective hierarchy in expansions of the real field by sequences

Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define $\mathbb N$. In particular, let $f\colon\mathbb R\to\mathbb R$ be such that $\lim_{x\to+\infty}f(x)=+\infty$, $f(x)=O(e^{x^N})$ as $x\to +\infty$ for some $N\in\mathbb N$, $(\mathbb R, +,\cdot,f)$ is o-minimal, and the expansion of $(\mathbb R,+,\cdot)$ by the set $\{\,f(k):k\in\mathbb{N}\,\}$ does not define $\mathbb N$. Then there exist $r>0$ and $P\in\mathbb R[x]$ such that $f(x)=e^{P(x)}(1+O(e^{-rx}))$ as $x\to +\infty$.