JEE Advanced 2018 Paper 1Question 1
For each positive integer $n$, let $$y_n=\frac{1}{n}\left((n+1)(n+2)\cdots(n+n)\right)^{\frac{1}{n}}.$$ For $x\in\mathbb{R}$, let $[x]$ be the greatest integer less than or equal to $x$. If $\lim_{n\to\infty}y_n=L$, then the value of $[L]$ is _____ .
Limits & Continuity
Answer$1$
Solution
$\ln L=\int_0^1\ln(1+x)\,dx=2\ln2-1$, so $L=4/e\approx1.47$ and $[L]=1$.