Let $f:(0,+\infty)$ be a differentiable function and $$\lim_{x \to +\infty} f'(x) = 0.$$ I need to show that $\lim_{x \to +\infty} \dfrac{f(x)}{x} = 0$.
My idea was to take a function $\tilde{f}:[0,+\infty) \to \mathbb{R}$ that extends $f$ continuously. Hence, taking a sequence of enclosing intervals $$[0, n_1] \subset [0, n_2] \subset \ldots \subset [0,n_k] \subset \ldots$$ we have, by mean value theorem, the existence of a sequence $\tilde{f}'(c_k)$, with $c_k \in [0,n_k]$, such that $$ \dfrac{\tilde{f}(n_k)}{n_k} = \left(\dfrac{\tilde{f}(n_k)}{n_k} - \dfrac{\tilde{f}(0)} {n_k} \right) + \dfrac{\tilde{f}(0)}{n_k} = \tilde{f}'(c_k) + \dfrac{\tilde{f}(0)}{n_k} \longrightarrow 0,$$ when $k \to +\infty$. Soon, the result would follow immediately.
My question is the following: using only the arguments of real analysis, how do I guarantee that there is this continuous extension $\tilde{f}$ of $f$?
