Question

x趋于0时，[(√1+x)+(√1-x)-2]/x^2=？

求解啊，要有步骤的，谢谢

Answer 1

(1+x)^(1/2)-1+(1-x)^(1/2)-1
=x/((1+x)^(1/2)+1)-x/((1-x)^(1/2)+1)
=x{1/((1+x)^(1/2))-1/((1-x)^(1/2)+1)}
又
1/((1+x)^(1/2)+1)-1/((1-x)^(1/2)+1)
={(1-x)^(1/2)-(1+x)^(1/2)}/{((1+x)^(1/2)+1)((1-x)^(1/2)+1)}
=(1-x-1-x)/{((1+x)^(1/2)+1)((1-x)^(1/2)+1)((1-x)^(1/2)+(1+x)^(1/2))}
那么总起来有
(1+x)^(1/2)-1+(1-x)^(1/2)-1
=x*(-2x)/{((1+x)^(1/2)+1)((1-x)^(1/2)+1)((1-x)^(1/2)+(1+x)^(1/2))}
=-2x^2/{((1+x)^(1/2)+1)((1-x)^(1/2)+1)((1-x)^(1/2)+(1+x)^(1/2))}
上式再除x^2, 另x趋向于0，就有极限为-1/4