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The Stirling's approximation formula: n ! ∼ 2 π n ( n e ) n ,   where the sign ~ means that the two quantities are asymptotic: their ratio tends to 1 as n tends to infinity.   This formula can be used to solve limits of sequences involving factorial.   Practice problems: 1. The following problem is asked in CSIR NET 2020(TN-PY) Find the limit: $\lim_{n \to \infty} \frac{((n+1)(n+2)\cdots(n+n))^{\frac{1}{n}}}{n}$  Note: Here we use ${n}^\frac{1}{n} \to 1.$ (Prove it also.) 2. $\lim_{n \to \infty}( \frac{(n!)}{(kn)^n})^{\frac{1}{n}}$ where $k \neq 0$