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.

Rewrite, $\lim_{n \to \infty}(\sqrt[3]{n^3+\sqrt{n}}-\sqrt[3]{n^3-1})\cdot \sqrt{(3n^3+1)}$ 
as $n\left(\left(1+\frac{1}{n^{\frac{5}{2}}} \right)^{\frac{1}{3}} -\left(1-\frac{1}{n^3} \right)^{\frac{1}{3}} \right)\times \sqrt{3}n^{\frac{3}{2}}\left(1+\frac{1}{3n^3}\right)^{\frac{1}{2}}$
Now use the Binomial theorem for fractional powers.
