Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{rl}& \underset{n\to \infty }{\lim }\prod _{r=3}^{\infty }\frac{r^3-8}{r^3+8}=\left(\frac{3^3-8}{3^3+8}\right)\left(\frac{4^3-8}{4^3+8}\right)\\ & \text{......... }\left(\frac{n^3-8}{n^3+8}\right)\end{array}$ $\begin{array}{r}={\lim }_{n\to \infty }\left[\frac{(3-2)\left(3^2+2^2+3.2\right)}{(3+2)\left(3^2+2^2-3.2\right)}\right]\left[\frac{(4-2)\left(4^2+2+4.2\right)}{(4+2)\left(4^2+2^2-4.2\right)}\right]\\ \dots \dots \dots ..\left[\frac{(n-2)\left(n^2+2^n+n.2\right)}{(n+2)\left(n^2+2^2-n.2\right)}\right]\end{array}$ $\begin{array}{c}=\underset{n\to \infty }{\lim }\left[\frac{(3-2)(4-2)\dots \dots \dots (n-2)}{(3+2)(4+2)\dots \dots \dots (n+2)}\right]\\ \left[\frac{\left(3^2+2^2+3.2\right)\left(4^2+2^2+4.2\right)\dots \dots .\left(n^2+2^2+n.2\right)}{\left(3^2+2^2-3.2\right)\left(4^2+2^2-4.2\right)\dots \dots ..\left(n^2+2^2-n.2\right)}\right]\end{array}$ $=\left[\frac{1\cdot 2\cdot \mathrm{3.4.5.6}\dots \dots \dots .}{5\cdot \mathrm{6.7.8}\dots \dots \dots ..}\right]\left[\frac{19\cdot 28\cdot \mathrm{39.52.63}\dots \dots \dots ..}{7\cdot 12\cdot \mathrm{19.28.39.52}\dots \dots ..}\right]$ $=\frac{2}{7}$