the limit of a probability density function I have one maths challenge which I could not resolve. The question is the following. Let f(x) denotes the probability density of a continuous RV., X. I want to know the limit f(x) as x approaches to either -∞ or +∞. It might seem silly, but I have do idea how to do that. 
 A: It is not necessary that a density have any limit as $x\to\infty$ or $x\to -\infty$.  The sole criteria for a function $f$ to be a density are $f\ge 0$ a.e. and that
$$\int_{-\infty}^\infty f(x)\mbox{d}x = 1.$$
Such a function could have high, narrow spikes at either or both ends.  It is, in fact, possible to have
$$\limsup_{n\to\infty} f(x) = \infty$$ and 
$$\limsup_{n\to-\infty} f(x) = \infty.$$
Try to construct such an example. It is not terribly hard.
A: Assuming that this question arose in an undergraduate (non-measure-theoretic)
course in probability, the answer that the OP (or perhaps the OP's instructor) 
might have been looking for 
might be that 
$$\lim_{x\to\infty} f(x) = \lim_{x\to -\infty} f(x) = 0.$$
The rationale would be that if $f(x)$ were converging to a positive number $c$
as $x \to \infty$, then
$\int_{-\infty}^{\infty}f(x)\,\mathrm dx$ would diverge. In more detail
for the OP's benefit, there would exist an $x_0$ such that for all $x > x_0$,
$|f(x) - c| < \frac{c}{2} \Rightarrow f(x) > \frac{c}{2}$, and so
$$\int_{-\infty}^{\infty}f(x)\,\mathrm dx 
\geq \int_{x_0}^{\infty}f(x)\,\mathrm dx
\geq \int_{x_0}^{\infty}\frac{c}{2}\,\mathrm dx.$$
Indeed, not only must $f(x)$ converge to $0$ as $x \to \pm\infty$
but the convergence
must be faster than $|x|^{-1}$.
Of course, since $f(x)$ can be viewed as a representative member of a
class of density functions that differ only on a set of measure zero,
one can find other functions in this class for which 
$\limsup f(x) = \infty$, e.g.
$$fx) = \begin{cases}
\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right),
&x ~\text{is irrational},\\
|x|, &x ~\text{is rational.}
\end{cases}$$
