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integral and limit f_n(x)= n^2x or 3/n

Source: Spanish Mathematical Olympiad 1981 P5

December 23, 2022
calculusintegrationalgebralimitfunction

Problem Statement

Given a nonzero natural number nn, let fnf_n be the function of the closed interval [0,1][0, 1] in RR defined like this: fn(x)={n2x,   if   0≤x<1/n3/n,   if   1/n≤x≤1f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases} a) Represent the function graphically. b) Calculate An=∫01fn(x)dxA_n =\int_0^1 f_n(x) dx. c) Find, if it exists, lim⁡n→∞An\lim_{n\to \infty} A_n .
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