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.

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.

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

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 relativeĀ 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).