<s>[INST] Limit of a Sequence involving 3rd root I'm not finding any way to simplify and solve the following limit:
$$
\lim_{n \to \infty} \sqrt{n^2+n+1}-\sqrt[3]{n^3+n^2+n+1}
$$
I've tried multiplying by the conjugate, but this give a more complex limit.
 [/INST] Notice the $n^{th}$ term of the sequence can be rewritten as
$$
  \sqrt{n^2+n+1} - \sqrt[3]{n^3 + n^2 + n + 1}
= \sqrt{\frac{n^3-1}{n-1}}-\sqrt[3]{\frac{n^4-1}{n-1}}
= n \left[\sqrt{\frac{1-n^{-3}}{1-n^{-1}}} - \sqrt[3]{\frac{1-n^{-4}}{1-n^{-1}}}\right]
$$
When one expand what's in the square/cubic roots in the rightmost expression
as a series in $n^{-1}$, the numerator contributes nothing up to order $O(n^{-3})$. 
For the denominators, we have
$$\frac{1}{\sqrt{1-n^{-1}}} \sim 1 + \frac12 n^{-1} + O(n^{-2})
\quad\text{ AND }\quad
\frac{1}{\sqrt[3]{1-n^{-1}}} \sim 1 + \frac13 n^{-1} + O(n^{-2})$$
As a result, the $n^{th}$ term has the dependence:
$$n \left[
\left(1 + \frac12 n^{-1} + O(n^{-2})\right) - 
\left(1 + \frac13 n^{-1} + O(n^{-2})\right)
\right]
= \frac{1}{6} + O(n^{-1})
$$
This immediately tells us the limit is $\frac16$ as $n \to \infty$.
</s>