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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?

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    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.2012-02-02

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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,

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    We don't use integer part in calculus or analysis context? Really?2012-02-02
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    Limits are sometimes important in number theory, too.2012-02-02
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    I looked at the question as an exercise in a course. Don't you agree now?2012-02-02
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    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)2012-02-02
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    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.2012-02-02
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    @Jonas : I added the "*usually*" so that my point is more clear.2012-02-03
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    I really do not see the problem... $\lfloor x \rfloor$ means the floor function (rounded down)$\\$ $\lceil x \rceil$ means the ceiling function (rounded up) $\\$ $\{x\}$ means the fractional part of x. \\ In almost all cases [] are normal bracets, used to make the math more readable. Eg using 5 nested () normal parenthesis would often look bad. A bit of thinking is also adviced when reading mathematics =) Cheers2012-02-03
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    @N3buchadnezzar : There's no "problem", there's just a question. =P2012-02-03
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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: ().

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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) $.

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The bracket function is denoted by [ ], and is defined as [x] is equal to the largest integer that is equal or less then x For Example (1) [5.5]=5 (2) [-0.1]=-1 (3) [-1.9]=-2

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The bracket function $[x]$ means the greatest value of $x$. For example: If we put $x = 5.5$ then the value returned by this function will be $x = 5$. Let's see $[5.5] = 5$ because the greatest integer is $5$.