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How do I show that this limit is zero?

$$\lim_{n \to \infty} \sqrt[8]{n^2+1} - \sqrt[4]{n+1} = 0$$

I've done the multiply by conjugates thing, which seems to lead nowhere:

$$\lim_{n \to \infty} (\sqrt[8]{n^2+1} - \sqrt[4]{n+1}) (\frac{\sqrt[8]{n^2+1} + \sqrt[4]{n+1}}{\sqrt[8]{n^2+1} + \sqrt[4]{n+1}})$$

$$=$$

$$\lim_{n \to \infty} \frac{\sqrt[4]{n^2+1} + \sqrt[2]{n+1}}{\sqrt[8]{n^2+1} + \sqrt[4]{n+1}}$$

I also considered using the squeeze theorem, as $\sqrt[8]{n^2+1} - \sqrt[4]{n+1} \leq \sqrt[8]{n^2+1} - \sqrt[4]{n}$, but I'm not sure how to bound it from below.

What's the correct approach to solving this limit?

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    $\begingroup$ Extract $\sqrt[4]{n}$ from both terms to get $\sqrt[4]{n}(\sqrt[8]{1 + n^{-2}} - \sqrt[4]{1 + n^{-1}}) = \sqrt[4]{n}(1 + O(n^{-2}) - 1 - O(n^{-1})) = O(n^{-3/4})$. $\endgroup$ Oct 5, 2013 at 10:40

2 Answers 2

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From identity $$\sqrt[8]a-\sqrt[4]b=\frac{a-b^2}{(\sqrt[8]{a}+\sqrt[4]{b})(\sqrt[4]{a}+\sqrt{b})(\sqrt{a}+b)}$$ for $a=n^2+1$ and $b=n+1$ we have that $$\lim_{n\to\infty}\sqrt[8]{n^2+1}-\sqrt[4]{n+1}=$$

$$=\lim_{n\to\infty}\frac{1}{\sqrt[8]{n^2+1}+\sqrt[4]{n+1}}\frac{1}{\sqrt[4]{n^2+1}+\sqrt{n+1}}\frac{n^2+1-{(n+1)^2}}{\sqrt{n^2+1}+n+1}=$$

$$=\lim_{n\to\infty}\frac{1}{\sqrt[8]{n^2+1}+\sqrt[4]{n+1}}\frac{1}{\sqrt[4]{n^2+1}+\sqrt{n+1}}\frac{-2}{\sqrt{1+1/n^2}+1+1/n}=0\cdot0\cdot(-2)=0$$

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  • $\begingroup$ Isn't the numerator of the last fraction $-2n-2$? $\endgroup$
    – kennysong
    Oct 5, 2013 at 13:30
  • $\begingroup$ No it is not it is $-2n$ $\endgroup$
    – Adi Dani
    Oct 5, 2013 at 15:14
  • $\begingroup$ Ah you're right! But then how do you get rid of the $n$? $\endgroup$
    – kennysong
    Oct 6, 2013 at 7:39
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    $\begingroup$ Last fraction is canceled by $n$ $\endgroup$
    – Adi Dani
    Oct 6, 2013 at 8:51
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Putting $\displaystyle n=\frac1{h^4},$

$$\lim_{n \to \infty} \sqrt[8]{n^2+1} - \sqrt[4]{n+1}$$

$$=\lim_{h\to0}\frac{(1+h^8)^{\frac18}-(1+h^4)^{\frac14}}h$$

Using Generalized Binomial Theorem

$$\lim_{h\to0}\frac{(1+h^8)^{\frac18}-(1+h^4)^{\frac14}}h=\lim_{h\to0}\frac{1+\frac{h^8}8+O(h^{16})-\{1+\frac{h^4}4+O(h^8)\}}h=\lim_{h\to0}\frac{-\frac{h^4}4+O(h^8)}h=0$$

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