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Suppose $f:[0,\infty)$ $\rightarrow$ R and has continuous $f'(x)$ and $f”(x) $ .

$f(x) $$\rightarrow $0 as $x$ $\rightarrow$ $\infty$. Given $f'(x)\rightarrow b$ as $x \rightarrow \infty$

show that $b=0$

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Get stuck in the first place, but I tried to use Mean Value Theorem and then connected it to the limits.

Any hint?

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**Hint**: consider

$$ \lim\limits_{x \rightarrow \infty} \frac{f(x)}{x} $$

**Further hint**: use L’Hospital’s rule.

**Notice** that to do that, you need the limit of $ f'(x) $ to exist. The counterexample has been mentioned

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