Asymptotic behavior

I very much dislike the "Big Oh" notation. It just doesn't stick in my mind. Suppose $f$ is a continuous function and $f \in \text{O}( 1/|x|^{1+\epsilon})$ when $|x| \rightarrow \infty$ and for $0< \epsilon < 1$. Does this mean that $$\int_{-\infty}^\infty |f(x)|\cdot |x|^\epsilon \; dx < \infty ?$$

$$\int_1^{\infty} |f(x)| |x|^{\epsilon} \, dx \le \int_1^{\infty} \frac{C}{|x|} \, dx = \infty$$
for some constant $C$.
• Example. If $f(x)=|x|^{-1-\epsilon}$ when $|x|\geq 1.$ – DanielWainfleet Apr 9 '16 at 19:40