有这样一道题,先计算再求值[(x^2-1)*1/(x^2-2x)+1/(2x-x^2)÷1/x-1]*(x-2)/(x-1),其中x=209,小明误把x=209看成2009,但答案相同,为什么

Question

有这样一道题,先计算再求值[(x^2-1)*1/(x^2-2x)+1/(2x-x^2)÷1/x-1]*(x-2)/(x-1),其中x=209,小明误把x=209看成2009,但答案相同,为什么

Answer 1

[(x²-1)·1/(x²-2x)+1/(2x-x²)÷1/(x-1)]·(x-2)/(x-1)
=[(x²-1)/(x²-2x)-1/(x²-2x)]·(x-1)}·(x-2)/(x-1)
=(x²-1-x+1)/(x²-2x)·(x-2)(x-1)
=(x²-x)/(x²-2x)·(x-2)/(x-1)
=x(x-1)/[x(x-2)]·(x-2)/(x-1)
=1
因为化简结果和x无关,所以无论x取何值,结果都为1