Question: > 
> Let $f \geq 0$ be an integrable function on $[0,\infty)$ such that $f(0)=0$ and $f'(0)$ is finite. Show that
> $\displaystyle \int^\infty\_0 \dfrac{f(x)}{x} dx < \infty$.
> 
> 
> 

This is my assignment for Lebesgue integral. Based on what given, I know that $f$ is finite on $ [0,\infty)$ and $\lim\limits\_{x\to 0}\dfrac{f(x)}{x} < \infty$.

I am thinking of approximating $\dfrac{f(x)}{x}$ by a sequence of increasing non-negative function that converges pointwise to $\dfrac{f(x)}{x}$ and use the Monotone Convergence Theorem (Levi's Theorem) but things just do not seem right. Can you guys please help me with this problem? Any idea or hint is appreciated.

Thank you!

Answer: 
Hint: There exist constants $C$ and $\delta$ such that $|f(x)| \le C x$ for $0 \le x \le \delta$. Consider the integrals over $[0,\delta]$ and $[\delta, \infty)$ separately.
The only issue was near zero for $\int^\infty\_1\frac{f(x)}{x}\,dx\leq\int^\infty\_0f<\infty$.

Since $f'(0)$ is finite, and $f(0)=0$, there is $\delta<0$ such that $|f(x)/x|\leq \frac{2|f'(0)|}{2}$ for all $0<x<\delta$. The rest should follow.