# $\int _0^\infty f(x)$ exists and $f(x)$ is differentiable then $\lim _{x \to \infty} f '(x)$ exists. Counter example of this statement.

Can anyone give me a counter example of the statement

If $$\int_0^\infty f(x)$$ exists and $$f(x)$$ is differentiable then $$\lim _{x \to \infty} f'(x)$$ exists.

My attempt: I have thought one. First I draw $$1/x^2$$ in the first quadrant and $$-1/x^2$$ in the fourth quadrant. The area under the following curves are finite.

1) $$1/x^2$$

2) $$-1/x^2$$

3) $$x= 1$$.

Now I have drawn infinite number of $$y = x+c$$ at equal distances in that region. Then I joined those infinite lines by some smooth curve so that the curve remains differentiable. Now I think this function can be a counter example.

I am uploading one picture of my attempt. Can anyone please check it and if possible suggest me a better function. $$\frac{\sin\left(x^{10}\right)}{x^2}$$
(where the exponent of $$10$$ is simply to make sure our function oscillates fast enough).
• Actually simply $\sin (x^2)$ suffices. – Szeto Nov 22 '18 at 5:24