Question 41
Let $f : [0, \infty) \to \mathbb{R}$ be a differentiable function, with continuous derivative, such that $f(0) \gt 0$, $f'(x) + (f(x))^2 \gt 0$, for any $x \gt 0$ and
$$
\lim_{x \to \infty} [f'(x) + (f(x))^2] = 0.
$$
Show that $\lim_{x \to \infty} f(x) = 0$.
$$
\lim_{x \to \infty} [f'(x) + (f(x))^2] = 0.
$$
Show that $\lim_{x \to \infty} f(x) = 0$.