The behavior of a density function at infinity Give $f$ the density function of a random variable. Does it follow that 
$$\lim_{x\rightarrow \pm\infty}xf(x)=0?$$
I really appreciate it if someone can give me a clue.

It does not follow that $\lim_{x\to\infty}f(x)=0$, though counterexamples are perhaps somewhat unnatural. 
For $x>0$, let $f(x)$ have a triangular "bump" of height say $1$ and base $\frac{2}{2^n}$ at every positive integer $n$, and let $f(x)=0$ elsewhere.  So the curve $y=f(x)$ climbs in a straight line from $(n-\frac{1}{2^n},0)$ to
$(n,1)$, then falls in a straight line to $(n+\frac{1}{2^n},0)$.
Then $\int_{-\infty}^\infty f(x)\,dx=1$, and $f(x)$ is non-negative. Note that $xf(x)$ is very large when $x$ is a large positive integer. 
This example can be "smoothed out" in various ways.  The behaviour of $xf(x)$ at integers can be assigned essentially freely.  
