Let $f: [0, +\infty)\rightarrow[0, +\infty)$ if differentiable function and $f'>0$ on all $[0, +\infty)$.

Prove that for all $\epsilon>0$:

$$\lim_{t\rightarrow+\infty} \frac{1}{t^2} \int_0^t \frac{f^{1+\epsilon}(x)}{f'(x)} dx=+\infty$$

My attempt:

It is true that $t^2$ and $\int_0^t \frac{f^{1+\epsilon}(x)}{f'(x)}dx$ are to $+\infty$ then $t$ to $+\infty$, so it is enough to prove that:

$$\lim_{t\rightarrow+\infty} \frac{f^{1+\epsilon}(t)}{t\cdot f'(t)}=+\infty$$

But it is not true! So, what I can do another?



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