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.

Calculate: $\lim _{n\rightarrow \infty }{n}^{3/2}[\sqrt {{n}^{3}+3}-\sqrt {{n}^{3} -3}]$


What do the brackets mean? I know sometimes they are used to denote a function that returns only the integer part of a number, like $f(x) = [x]$ has values of $0$ on $(0,1)$ and then jumps to $1$ on [1,2) and then $2$ on $[2,3)$ and so on... Is this what is meant here?

share|improve this question
1  
If the square brackets meant integer part, as you speculate, the answer would be fairly easily $0$. I am reasonably confident that you are expected to think of them as a variant of ordinary parentheses. If you want, you can cover all possible bases by adding a remark about what happens if one interprets the square brackets as meaning integer part. –  André Nicolas Feb 2 '12 at 20:05
add comment

3 Answers 3

up vote 2 down vote accepted

Here the brackets are equivalent to $($ $)$. I am saying so because we don't usually use the integer part function in a calculus or analysis context. If you are in a number theory context, then those might mean the integer part function, but then I don't see why you would be computing this limit.

Hope that helps,

share|improve this answer
3  
We don't use integer part in calculus or analysis context? Really? –  Aryabhata Feb 2 '12 at 20:02
    
Limits are sometimes important in number theory, too. –  Jonas Meyer Feb 2 '12 at 20:16
    
I looked at the question as an exercise in a course. Don't you agree now? –  Patrick Da Silva Feb 2 '12 at 20:21
    
thanks, I was getting no where with integer-part. To all you professors and teachers out there, remember that a small typo or brackets instead of regular parentheses can mean hours of confusion for your students. (I took this from a previous test so maybe the professor was there to explain that these are just ordinary parenthesis and were miss-typed for some reason) –  nofe Feb 2 '12 at 20:24
    
Patrick: I do agree that it is equivalent to $()$ in this context. I don't agree that we don't use integer part in calculus or analysis context. I have personally assigned limit problems that involve the floor function, and they show up on this site sometimes too (many of them likely exercises in courses). @nofe: Thank you for the advice. I prefer $\lfloor x\rfloor$ to $[x]$ anyway for the floor function. –  Jonas Meyer Feb 2 '12 at 22:32
show 3 more comments

It is unlikely to be integer part, as that would make the $n^{3/2}$ term irrelevant.

I am quite sure that it is the normal brackets: ().

share|improve this answer
add comment

Or, using the binomial theorem, ${n}^{3/2}(\sqrt {{n}^{3}+3}-\sqrt {{n}^{3}-3}) = n^3(\sqrt {1+3/n^3}-\sqrt {1-3/n^3}) $ $ = n^3((1 + 3/(2n^3) + O(1/n^6)) - (1 - 3/(2n^3) + O(1/n^6)) $ $ n^3(3/n^3 + O(1/n^6) = 3 + O(1/n^3) $.

share|improve this answer
add comment

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.