Evaluation of limits: $\lim n \cdot \exp( f(nx) )$ I have two similar (from my point of view) limits:
$$\lim_{n \rightarrow \infty} x \sqrt n \exp(-\sqrt {nx})$$
$$\lim_{n \rightarrow \infty} n \exp(-n \cdot |x|))$$
How to deal with such limits? Is there any general procedure for solving limit like $\lim n \cdot \exp( f(nx) )$ where $f(\cdot)$ is some function.
 A: Not really any general procedure in the absence of knowledge of $f$.  In your two cases, the first is $0$ when $x \ge 0$ and indeterminate (modulus $\infty$) when $x \lt 0$.  The second one is $0$ when $x \ne 0$ and $\infty$ when $x=0$. 
The solutions above are based on the fact that the behavior of $y \, e^{a y}$ depends on the sign of $a$.  For example
$$\lim_{y \to \infty} \frac{y}{e^{-a y}} = \lim_{y \to \infty}  \frac{1}{-a e^{-a y}}$$
by L'Hopital's rule.  If $a \ge 0$, the limit is not finite.  The $a \lt 0$, the limit is zero.
A: Since it is not  $ \ x \ $ , but rather $ \ n \ $ for which the "limit at infinity" is being taken, the problem is describing two sequences of functions, each of which has some interesting peculiarities.
The second sequence describes functions $ \ f(x, n) \ = \ n \cdot e^{-n|x|} \ , $ which have even symmetry, horizontal asymptotes of  $ \ y = 0 \ $ , and maximum values $ \ f(0, n) = n \ . $  The areas under the curves are 
$$A(n) \ = \ 2 \  \int_0^{\infty} n \cdot e^{-nx} \ dx \ = \ 2 \ . $$
We would now consider the meaning of the limit $ \ n \ \rightarrow \infty \ $ for this:  the curve comes down faster toward the horizontal asymptote and the area under the curve tends to zero, but the peak at $ \ x = 0 \ $ grows without limit.  The limiting function would be zero everywhere, except at $ \ x = 0 \ $ , where it is undefined.  (Because the limiting area is 2, this is a relative of the "Dirac delta" function.  I'm not much of an analyst, but I believe this is an example of a function which is not uniformly convergent...)
Here's a graph for the functions with $ \ n = 1 , 3 , \  \text{and} \ 10 \ $ (red, green and blue, respectively).

Returning to the first sequence, the functions $ \ g(x,n) = x \cdot \sqrt{n} \cdot e^{-\sqrt{nx}} \  $ are defined for $ \ x \ge 0 \ , $ again have horizontal asymptotes $ \ y = 0 \ , $ values $ \ f(0) = 0 \ , $ and maximum values $ \ f(\frac{4}{n}) = \frac{4}{\sqrt{n}} \cdot e^{-2} \ . $  The limiting area under the curves is zero.  So this set of curves flattens toward the non-negative $ \ x-$axis as  $ \ n \rightarrow \infty  \ , $ but not uniformly; the limit might be described as $ \ y = 0 \ , \ x \ge 0 $ .  
Here is a graph for the functions with $ \ n = 1 , 3 , 10 \  \text{and} \ 100 \ $ (blue, green, red and black, respectively).

