Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm having a hard time characterising the behavior of the following expression:

$$\lim_{n\rightarrow\infty}\left(\frac{2\sqrt{a(a+b/(\sqrt{n}+\epsilon))}}{2a+b/(\sqrt{n}+\epsilon)}\right)^{\frac{n}{2}}$$

with the following constraints on the parameters: $0<b<a<\infty$, and $\epsilon\in\mathbb{R}$. I am interested in the following:

  1. for $\epsilon>0$, does this limit go to zero or does it go to some constant $C$? If it can both go to zero or to some constant $C>0$, what are the conditions on the value of $\epsilon$ as a function of $a$ and $b$ which leads to these outcomes, if any?
  2. for $\epsilon<0$, does it always go to some constant $C<1$, or can it go to 1 for some $\epsilon$, if it's a function of $a$ and $b$?
  3. what happens to this limit when $\epsilon=0$?
share|improve this question

1 Answer 1

up vote 7 down vote accepted

Claim: The limit is $\exp(-b^2/(16a^2))$, irrespective of $\epsilon$.

Proof: Let $x=b/(2a(\sqrt{n}+\epsilon))$, then one asks for the behaviour of $$ K_n=\left(\frac{1+2x}{(1+x)^2}\right)^{n/4} $$ when $n\to\infty$, with $x\to0$. But $\frac{1+2x}{(1+x)^2}=1-\frac{x^2}{(1+x)^2}=1-x^2+o(x^2)$ and $x^2\sim c^2/n$ with $c=b/(2a)$, hence $$ K_n=\left(1-\frac{c^2}n+o\left(\frac1n\right)\right)^{n/4}\longrightarrow\exp\left(-\frac{c^2}4\right). $$

share|improve this answer
    
Thanks, very nice proof! –  M.B.M. Oct 31 '11 at 22:27
    
I posted another limit question, which is sort of related to this one: math.stackexchange.com/questions/77655/… –  M.B.M. Oct 31 '11 at 23:05

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.