Does there exist a strictly monotone function $f\colon \Bbb R\to\Bbb R$ that is $\mathcal{C}^\infty$, and $\lim\limits_{x\to+\infty} f(x) = 0$, but
$$\lim_{x\to+\infty} f^\prime(x) $$
and $\lim\limits_{x\to-\infty} f^\prime(x)$ do not exist?
en, counterexamples can be found, according to 'Obvious' theorems that are actually false
Thank you very much!
$\newcommand{\R}{\mathbf{R}}$Let $\phi:\R \to \R$ be a smooth, non-negative function having a thin "bump" of height $1$ near each integer, and such that the bumps have finite total area. Define $$ g(x) = \phi(x) + \frac{1}{1 + x^{2}},\qquad f(x) = -\int_{x}^{\infty} g(t)\, dt. $$ Since $g = f'$ is smooth and strictly positive, $f$ is strictly increasing. By construction, $g(x) = f'(x)$ has no limit as $x \to \infty$
