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limits of function on two variables

Source: France 1991 P2

May 14, 2021
functionlimitalgebra

Problem Statement

For each n∈Nn\in\mathbb N, the function fnf_n is defined on real numbers x≥nx\ge n by fn(x)=x−n+x−n+1+…+x+n−(2n+1)x.f_n(x)=\sqrt{x-n}+\sqrt{x-n+1}+\ldots+\sqrt{x+n}-(2n+1)\sqrt x.(a) If nn is fixed, prove that lim⁡x→+∞fn(x)=0\lim_{x\to+\infty}f_n(x)=0. (b) Find the limit of fn(n)f_n(n) as n→+∞n\to+\infty.
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