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Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$

Could someone help me through this problem? Calculate $\displaystyle\lim_{n \to{+}\infty}{(\sqrt{n^{2}+n}-n)}$

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marked as duplicate by Pedro Tamaroff, cardinal, Aryabhata, Zev Chonoles Apr 25 '12 at 5:21

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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What have you tried so far? Do you know any algebraic techniques to rewrite expressions that are written as differences where at least one of the terms is a square root? –  Michael Joyce Apr 24 '12 at 23:13
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@Peter: I don't think that thread does a very good way of avoiding duplicates, as most people who are asking about problems such as the one in this thread are not yet prepared to comprehend the general solution. –  Michael Joyce Apr 24 '12 at 23:49
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Yes, "abstract duplicate" has little meaning for students whose background is mainly computational. –  André Nicolas Apr 25 '12 at 0:41

3 Answers 3

up vote 4 down vote accepted

We have:

$$\sqrt{n^{2}+n}-n=\frac{(\sqrt{n^{2}+n}-n)(\sqrt{n^{2}+n}+n)}{\sqrt{n^{2}+n}+n}=\frac{n}{\sqrt{n^{2}+n}+n}$$ Therefore:

$$\sqrt{n^{2}+n}-n=\frac{1}{\sqrt{1+\frac{1}{n}}+1}$$

And since: $\lim\limits_{n\to +\infty}\frac{1}{n}=0$

It follows that:

$$\boxed{\,\,\lim\limits_{n\to +\infty}(\sqrt{n^{2}+n}-n)=\dfrac{1}{2}\,\,}$$

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Guide: Rationalize,

$$\left(\sqrt{n^2+n}-\sqrt{n^2}\right)\cdot \frac{\sqrt{n^2+n}+\sqrt{n^2}}{\sqrt{n^2+n}+\sqrt{n^2}}=\frac{n}{\sqrt{n^2+n}+\sqrt{n^2}}$$

Now divide numerator and denominator by $n$. Remember $\frac{1}{n}\sqrt{\square}=\sqrt{\frac{1}{n^2}\square}$.

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Here's an answer that is probably not within the intended scope but it's nice anyway...

Let $x=1/n$. Then $$ \lim_{n\to{+}\infty}{\sqrt{n^{2}+n}-n} = \lim_{x\to0}{\sqrt{\frac1{x^2}+\frac1x}-\frac1x} = \lim_{x\to0}{\sqrt{\frac{1+x}{x^2}}-\frac1x} = \lim_{x\to0}{\frac{\sqrt{1+x}}{x}-\frac1x}= \lim_{x\to0}{\frac{\sqrt{1+x}-1}{x-0}} = f'(0) = \frac12 $$ for $f(x)=\sqrt{1+x}$.

(There's a small technicality that actually $x\to0^+$ but let's overlook that.)

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You can just take $x \rightarrow 0^+$ the whole way through and conclude as you do, since if $\lim_{x \rightarrow 0} g(x)$ exists, then so does $\lim_{x \rightarrow 0^+} g(x)$ and the two limits are equal. (Minor point: you have a $1/n$ that should be a $1/x$.) –  Michael Joyce Apr 24 '12 at 23:46
    
@MichaelJoyce, that's the small technicality I meant. And thanks for finding that typo. –  lhf Apr 24 '12 at 23:50

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