<s>[INST] Limit of a sequence involving roots How can I find the limit of the following sequence:
$b_n = \sqrt[4]{n^4+n^3} - \sqrt[4]{n^4+1}$?
I succeeded in finding the limits of other sequences involving roots using the binomic formulas but I don't know what to apply to fourth roots.
Thanks for your help!
 [/INST] This calls for multiplication with the conjugate: $\sqrt a-\sqrt b=\frac{a-b}{\sqrt a+\sqrt b}$.
$$ \begin{align}b_n&=\sqrt[4]{n^4+n^3}-\sqrt[4]{n^4+1}\\
&=\frac{\sqrt{n^4+n^3}-\sqrt{n^4+1}}{\sqrt[4]{n^4+n^3}+\sqrt[4]{n^4+1}}\\
&=\frac{(n^4+n^3)-(n^4+1)}{(\sqrt[4]{n^4+n^3}+\sqrt[4]{n^4+1})(\sqrt{n^4+n^3}+\sqrt{n^4+1})}\\
&=\frac{n^3}{(\sqrt[4]{n^4+n^3}+\sqrt[4]{n^4+1})(\sqrt{n^4+n^3}+\sqrt{n^4+1})}\\
&=\frac{1}{(\sqrt[4]{1+n^{-1}}+\sqrt[4]{1+n^{-4}})(\sqrt{1+n^{-1}}+\sqrt{1+n^{-4}})}\\&\to \frac1{(1+1)(1+1)}=\frac14\end{align}$$
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