Limit of: $\lim_{n \to \infty}(\sqrt[3]{n^3+\sqrt{n}}-\sqrt[3]{n^3-1})\cdot \sqrt{(3n^3+1)}$ I want to find the limit of: $$\lim_{n \to \infty}(\sqrt[3]{n^3+\sqrt{n}}-\sqrt[3]{n^3-1})\cdot \sqrt{(3n^3+1)}$$
I tried expanding it by $$ \frac{(n^3+n^{1/2})^{1/3}+(n^3-1)^{1/3}}{(n^3+n^{1/2})^{1/3}+(n^3-1)^{1/3}} $$
but it didn't help much. Wolfram says the answer is $\frac{3^{1/2}}{3}$.
Any help would be greatly appreciated.

With the expansions
$$
(n^3+\sqrt{n})^{1/3}\sim n+(1/3)n^{-3/2}+...
$$
and
$$
(n^3-1)^{1/3}\sim n-(1/3)n^{-2}+...
$$
and
$$
(3n^3+1)^{1/2}\sim \sqrt{3}n^{3/2}+...
$$
as $n\to\infty$, the result follows immediately.
