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Find $\lim \limits_{x\to a_m}\Big(A_1A_2\cdots A_n\Big)$

Source: MTRP 2015 Class 11-Multiple Choice Question: Problem 3 :-

July 15, 2020
number theory

Problem Statement

If Ai=x−ai∣x−ai∣A_i=\frac{x-a_i}{|x-a_i|}, i=1,2,⋯ ,ni=1,2,\cdots,n for nn numbers a1<a2<⋯<am<⋯<an,a_1<a_2<\cdots<a_m<\cdots<a_n, then lim⁡x→am(A1A2⋯An)=?\lim \limits_{x\to a_m}\Big(A_1A_2\cdots A_n\Big)=?
[*] (−1)m−1(-1)^{m-1} [*] (−1)m(-1)^m [*] 11 [*] None of these
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