IITJEE Mains 2016 (offline)

This limit can be rewritten as $\displaystyle\lim_{n \to \infty} \left(\dfrac{(3n)!}{n!*n^{2n}}\right)^{\frac{1}{n}}$ .

Now by Stirling's approximation $k! \to \sqrt{2\pi k}\left(\dfrac{k}{e}\right)^{k}$ as $k \to \infty$ . Our limit then takes the form

$\displaystyle\lim_{n \to \infty} \left(\dfrac{\sqrt{2\pi*3n}\left(\dfrac{3n}{e}\right)^{3n}}{\sqrt{2\pi n}\left(\dfrac{n}{e}\right)^{n}*n^{2n}}\right)^{\frac{1}{n}} = \lim_{n \to \infty} \left(\sqrt{3} * \dfrac{3^{3n}}{e^{2n}}\right)^{\frac{1}{n}} = \lim_{n \to \infty} 3^{\frac{1}{2n}} \times \lim_{n \to \infty} \left( \left(\dfrac{27}{e^{2}}\right)^{n}\right)^{\frac{1}{n}} = \boxed{\dfrac{27}{e^{2}}}$ .