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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.

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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.

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  • $\begingroup$ I'm more interested in solution rather than answer. Could you please explain, how you solved? $\endgroup$ – Surpri Jul 3 '13 at 17:27
  • $\begingroup$ @Surpri: see edit above. $\endgroup$ – Ron Gordon Jul 3 '13 at 17:32
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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).

enter image description here

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).

enter image description here

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