$\underset{n\to \infty }{\lim }\prod _{r=2}^n\left(\frac{r^3+1}{r^3-1}\right)=$

$\underset{n\to \infty }{\lim }\prod _{r=2}^n\left(\frac{r+1}{r-1}\right)\prod _{r=2}^n\left(\frac{r^2-r+1}{r^2+r+1}\right)$

$=\underset{n\to \infty }{Lt}\frac{3}{1}·\frac{4}{2}·\frac{5}{3}····\frac{n+1}{n-1}·\frac{3}{7}·\frac{7}{13}····\frac{n^2-n+1}{n^2+n+1}$

$=\underset{n\to \infty }{Lt}\frac{1}{1·2}n(n+1)\frac{3}{n^2+n+1}$

$=\underset{n\to \infty }{Lt}\frac{3}{2}\frac{1+\frac{1}{n}}{1+\frac{1}{n}+\frac{1}{n^2}}=\frac{3}{2}$