How to solve the limit $\lim_{n \to \infty} \sqrt[8]{n^2+1} - \sqrt[4]{n+1}$?

Question

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?

Answer 1

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

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

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

Answer 2

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

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