Let $f$ and $g$ be two functions with derivatives in some interval containing $0$, where $g$ is positive. Also
Prove or dissprove:
1) $$\int_0^xf(t)dt=o\left(\int_0^xg(t)dt\right)$$ 2) $$f'(x)=o(g'(x))$$
Now considering the first, my reasoning is as follows:
Now the first member on the right will tend to $0$. Second will also seems to converge to $0$ (though I am unsure of that). And the limit should converge to $0$? I realize this is a very weak reasoning. How could I make it more precise?
Considering the second problem I am quite clueless though I am quite sure it should converge to $0$ :) Any hints?