$f\colon [0, \infty) \to \mathbb R$ is $C^2$ and $f(x) \to 0$ as $x \to \infty$. If $f'(x) \to b$ as $x \to \infty$, show that $b = 0$.
Suppose $f:[0,\infty]\rightarrow$ to R and has continuous $f'(x)$ and $f''(x)$ f(x) → 0 as x → ∞. Show that f′(x)→0 as x→∞