Prove that limit of function is equal to zero

Question

We have a function $f(x)$ defined on $ [0, +\infty)$.

It is known that $|f'(x)|<2$.

Prove that if $\int_0^{\infty} f(x)\,dx$ converges then $\lim_{x \to +\infty} f(x)=0$.

I don't really know how to use the fact about derivative, I would like to apply LHopital's rule in order to jump from integral to function and then to derivative, but it is not suitable here.

I think it's $x$ @JackD'Aurizio – Learnmore May 30 '17 at 15:59 — Learnmore, May 30 '17 at 15:59
can you see why this is the case? – Cato May 30 '17 at 16:03 — Cato, May 30 '17 at 16:03

score 1 · Accepted Answer · answered May 30 '17 at 16:03

1

Hint. Show that $|f'(x)|<2$ implies that $f$ is uniformly continuous in $[0, +\infty)$. Then take a look here: $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

answered May 30 '17 at 16:03

Robert Z

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Prove that limit of function is equal to zero

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