I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose
$$\begin{cases}
f \in \mathcal{C}(\mathbb{R}) \\
\dot{x} = f(x) \\
x(0) = 0 \\
f(0) = 0 \\
\end{cases}.
$$
Then,
$$
\lim_{\varepsilon \searrow...